Could dark energy be the effect of gravity at great distances? This may be a silly question, but is it possible that dark energy and gravity are related to each other? 
Space-time is deformed everywhere in space by objects with mass. The more massive the object, the more deformation in the local area, and, as a result, the more concentrated space-time is within that deformation. However, where is the "extra" space-time coming from to be able to form these super-concentrated regions of space-time? Is that what we observe as dark energy?
Is it possible that at great enough distances, gravity could be pulling space-time apart due to the lack of mass in the great voids of space? Not only that, but if there was enough mass in a part of space to have a little more gravity than the surrounding area, it could prevent that area of space from being "stretched out." Thus, might dark matter just be regions of space where space-time is denser than the surrounding regions?
I'm sure these aren't new ideas, but I'd like to know what premises and assumptions are incorrect.
 A: Dark energy, dark matter, and gravity are intimately entwined concepts, certainly. The question seems to blur dark matter and dark energy together or use the terms interchangeably, but they are separate theories attempting to reconcile separate discrepancies.
One facet contributing to the majesty of Einstein's Equations for gravity (see http://en.wikipedia.org/wiki/Einstein_field_equations, particularly the form given just before "Sign convention") are that they are all-inclusive- curvature is mass-energy, mass-energy is curvature. The equation begs to be enlarged and expanded to include the entire universe, and thus it is the foundation of modern cosmology.
As such, Einstein's Equations describe how the Universe should expand, deform, warp, or contract (if it had enough mass) given its content of mass and energy. Dark energy is part of the nascent attempt by theorists to reconcile what matter we can see with the largest-scale behavior as observed. Gravity as we are traditionally familiar with it is always attractive, so any collection of objects moving apart from each other should at least be slowing down, if not actually turning around and colliding. What we see on the grandest scales, though, is an acceleration. Something fundamental has to be wrong with the traditional picture, and dark energy is a catch-all term for attempts at the modification of Einstein's Equations themselves.
Dark matter, on the other hand, is a catch-all term for attempts to reconcile smaller-scale discrepancies between the observed behavior of objects and the theoretical predictions of their behavior assuming that every object was visible. This is where the alternate term "Missing Matter" comes from. Smaller objects, like galaxies and clusters of galaxies, seen to behave as if there were extra, invisible matter holding them together. It is a less-ambitious correction, since there is no attempt to argue that Einstein's Equations themselves need correction, we merely need some extra matter that doesn't produce any light on the right hand side in order to account for the extra curvature of spacetime inferred from the behavior of observed objects on the left.
I'm sorry if that's not as neat or tidy an answer as you were looking for. Given the deep philosophical implications of both sides of Einstein's Equations being both cause and effect, it is impossible to draw the same kind of distinction between gravity, dark matter, and dark energy as you could between the influences of, say, sunshine and heat from the Earth's core on ground temperatures. Therefore, for instance, there is no "extra" spacetime. Einstein himself discouraged the use of "deformation" and "warping" as they lead to the notion that there is some natural condition for spacetime and matter just messes it all up. Spacetime and matter are like the first two cards of a house of cards- they have to be taken together because either one by itself will just collapse.
A: This idea has been suggested. The April 2011 issue of Scientific American had an article that suggests voids as an alternative to dark energy.
Many cosmologists accept that, due to inflation in the Big bang, our visible universe is only a small part of a much larger Universe, and it is possible that the visible universe happens to be a bubble of slightly lesser density compared to the rest of the Universe. In that case we would experience gravitational attraction from outside our universe, pulling our universe apart, giving a similar effect to dark energy. 
The pattern of acceleration would be different though: with voids, the furthest galaxies would be more affected than closer ones; with dark energy we find the opposite: closer galaxies are going faster than expected. Unfortunately, recent Hubble data has measured the expansion speeds with higher precision and seems to rule out the voids idea.
A: While not quite the gigaparsec-scale void alternative covered by hdhondt, a somewhat similar idea to your question has been suggested by David Wiltshire and a few others as "Dark Energy without Dark Energy", or "Timescape Cosmology".
They posit that the apparent acceleration of Dark Energy is really just an observer effect from General Relativity, which hasn't been correctly accounted for.  Standard cosmology assumes large-scale homogeneity, or at least that both the observer (us) and targets (supernovae) are in approximately identical locations.  If we're at a less-than-average gravitational potential, though (eg. in a bubble wall), then light from deeper gravitational potentials will be redshifted.  Additionally, light from earlier in the universe will have somewhat less shift than expected, having spent more of it's time in the more-homogeneous early universe.
I don't have a great understanding of the details of timescape cosmology, but I find the suggestion of GR effects being more complicated than an assumed large-scale homogeneity to be alluring.
A: I agree.  If gravity puts wells in space-time fabric (like a ball on a trampoline), then is there a median rest level for the fabric?  Why would an object only put a well in the fabric and not a reactionary hill?  Shouldn't gravity have a balancing repulsive force (modeled as a hill in the fabric)?  This hill would be unnoticeable in a larger well, but additive to the outside of the greater well (like a galaxy well).  So a ring of repulsive force around the rim of a galaxy would hold it together and push the objects to a greater velocity.  And if the galaxies are accelerating away from each other, it is because when they were close together they shared a big well.  But as they expanded in an expanding ring (or sphere) the inner side of the ring would be able to form a crest/hill that would push out on the galaxies more than the outer hill would (or maybe there is no outer hill due to instantaneous creation of space-time at the expansion boundary?).  So the inner ring would be low-power at the start, high-power expansion near the middle, but eventually dissipate with long-time expansion like a wave.  So galaxies would have a repulsive affect on medium objects approaching from the outside: low-speed objects and not massive objects.  I think the black hole at the center of our galaxy would have a bulge/hill region where matter would pile up on the outer side of the bulge, and speed up on the inner side of the bulge.
