I'll start with evidence at the largest and smallest scales in the Universe.
British astrophysicist Fred Hoyle once said
A common sense interpretation of the facts suggests that a superintellect has monkeyed with physics, as well as with chemistry and biology, and that there are no blind forces worth speaking about in nature. The numbers one calculates from the facts seem to me so overwhelming as to put this conclusion almost beyond question.
This conclusion has only been strengthened as modern study has uncovered deeper workings of the Universe. While the Bible isn't a science textbook outlining the workings of the Universe, it tells us that it had a beginning. It was only in 1929 that Hubble made the discovery that let "modern" science catch up with what the Bible has said for thousands of years.
That said, there is significant evidence we can look at which support the fact that our Universe was designed, and fine tuned, by God.
A quick note regarding the form of some numbers below. You will see a number written as 10^##. This can be written out as a 1 with the ## number of 0's after it. For example, 10^5 is 100,000 and 10^10 is 10,000,000,000. So not double, but a much larger number. Keep that in mind.
If you would like an idea of just how unlikely some of the below odds are (like one in 10^55 for example), either leave a comment or check out my Prophecies of Jesus post.
Let's take a look at some of the fine tuned aspects of our Universe.
The Fundamental Forces
Strong Nuclear Force Constant
The strong nuclear force is the strongest of the fundamental forces. It acts on an incredibly small distance and is what holds neutrons and protons together in the nucleus of atoms. It's fine tuned in a way that holds nuclei together, but can't hold two protons together without a neutron present. The electromagnetic repulsive force from the two positive protons by themselves overwhelms the strong force.
If the strong force constant was larger, two protons could stay together by themselves. If this happened, all hydrogen would have been converted to helium in the big bang. No hydrogen, no water among other things. It would be impossible to sustain life.
If the strong force constant was smaller, it would affect the stability of the nucleus. All elements essential to life wouldn't be able to stay together. Even weaker, only hydrogen could exist. Again, it would be impossible to sustain life.
Weak Nuclear Force Constant
The weak nuclear force is responsible for radioactive decay of atoms. Like the strong force, it acts on small distances. It's fine tuned in a way that allows stars to gradually burn their hydrogen.
If the weak force constant was stronger, we would have the same over production of hydrogen into helium that we saw in the strong force section.
If the weak force constant was smaller, not enough helium would have formed. This would have made later, heavier element production impossible. Without these elements, life could not have formed.
The Electromagnetic Force Constant
The electromagnetic force is one of the two fundamental forces we have every day experience with. Maxwell's equations showed us this relationship and it transformed our entire world. It's also relevant to stellar fusion and chemistry.
If the electromagnetic force changed, its relationship with the strong force would be disrupted among other things.
If the electromagnetic force constant was higher, electrons would have too much energy and atom stability would be affected. Life would be impossible.
If the electromagnetic force constant was lower, the atomic bonds could not form. Thus, the complex molecules for life chemistry would not be possible.
The Gravitational Force Constant
The gravitational force is the other force we have every day experience with. Reconciling gravity at the quantum level is a current goal of modern physics. However, what we have learned about this force is how incredibly fine tuned it is for life. Incredibly, the gravitational constant, if moved by 1 part in 10^34 (1 with 34 zeroes after it), life would not be possible.
If the gravitational force constant was higher, stars would become too hot and burn out too quickly. Additionally, the force of gravity on any planet would be significantly too large to support life.
If the gravitational force was weaker, stars would not be able to fuse the heavier elements necessary for life.
Ratios
Number of Protons to Number of Electrons
The number of protons in the Universe is equal to the number of electrons to one part in 10^37!
This relationship is what yields our electrically neutral Universe. Not only that, but these particles with very different masses have exactly equal charges.
If this was imbalanced in either direction, electromagnetism would dominate gravity. Stars would then not be able to form and life would be impossible.
Electromagnetic Force Constant to the Gravitational Force Constant
This ratio is incredibly finely balanced. To within one part in 10^40! It is vital to star formation.
If increased even slightly, all stars would grow to be at least 40% more massive than our Sun. If that occured, stars would burn through their fuel supplies too quickly and irregularly for life to survive.
If decreased even slightly, stars would be at least 20% less massive than our Sun. The resulting end of life cycle for stars would be unable to produce the heavier elements for life.
Proton to Electron Mass
The relationship between proton and electron mass is perfect for forming molecules. If changed in either direction, chemical bonding would cease to be sufficient for life chemistry.
Neutron to Proton Mass
Physical systems in nature tend to seek out their lowest energy state when they're not being disturbed. The Jefferson Lab defines beta decay as
Beta decay is one process that unstable atoms can use to become more stable. There are two types of beta decay, beta-minus and beta-plus.
During beta-minus decay, a neutron in an atom's nucleus turns into a proton, an electron and an antineutrino.
If the neutron to proton mass were to favor the proton (by as little as 1% more) instead of the other way around and isolated protons would turn into neutrons. This would not allow hydrogen to form and stars would not be able to produce the complex elements necessary for life.
Furthermore, as the proton is the lightest baryon, it cannot decay when left alone. If they did, all elements would be unstable as the neutrons would still decay in the same way described.
Characteristics of the Universe
Expansion Rate of the Universe
If the expansion rate of the Universe was changed by one part in 10^55, life would not be here!
If increased, matter would expand too quickly to allow for star and galaxy formation. If decreased, the Universe would have collapsed back in on itself before stars formed.
Mass Density of the Universe
If the mass density of the Universe were to differ by one part in 10^59, life would not be here!
If increased, too much deuterium would have formed in the big bang. This would cause stars to burn too rapidly for life formation. If decreased, not enough helium would have been produced to allow for the necessary abundance of heavy elements needed for life.
Supernovae
When stars go supernova, it distributes all the heavy elements necessary for planet formation and life. These events are incredibly energetic and dangerous. If a supernova were to explode near a planet with life, there would be significant ramifications.
If too many supernovae occured, life would have likely been wiped out from the energetic explosions. If they were too infrequent, there would not be sufficient amounts of heavy elements for planet formation and life.
Particles and Molecules
Decay Rate of Beryllium-8
If the decay rate of Beryllium-8 all the elements for life would not be able to form.
If increased, when stars started fusing heavier elements they would catastrophically explode. If decreased, no element more massive than Beryllium would form. In either case, life would not be able to exist.
Polarity of the Water Molecule
Water is a polarized molecule. The University of Arizona explains this as
Water is a "polar" molecule, meaning that there is an uneven distribution of electron density. Water has a partial negative charge near the oxygen atom due the unshared pairs of electrons, and partial positive charges near the hydrogen atoms. An electrostatic attraction between the partial positive charge near the hydrogen atoms and the partial negative charge near the oxygen results in the formation of a hydrogen bond
...
The unique physical properties, including a high heat of vaporization, strong surface tension, high specific heat, and nearly universal solvent properties of water are also due to hydrogen bonding.
If this polarity were stronger, heat of vaporization and fusion would be too high for life. If weaker, the heat of vaporization and fusion would be too low for life.
Mass of the Down Quark
Quarks are particles that make up protons and neutrons. They come in multiple varieties. The proton has two up quarks and one down quark. If you were to decrease the mass of the down quark, protons would capture electrons near them. This would make the formation of life impossible.
As science continues to make discoveries of the inner workings of our Universe, we find them consistent with a pattern of unique and well placed "coincidences" again and again, at the smallest and largest scales.
The volume of finely tuned parameters which support life and the degree of fine tuning, in my opinion, are more than enough to claim as evidence for design.
The sheer scale, specificity, and application of Laws which remain consistent all support for this design to be intelligent.
How would you define "proved" for something like the existence of a being who created and exists outside of our Universe?
I would say greater evidence of Unintelligent Design, but hard to say there's no evidence, even if said evidence falls well short of being proof.