I have an apple tree that is developing a lot of branches close together after it lost half its height due to a deer attack. As the tree grows taller again, will the new branches eventually grow with more space between them?
No, they will not.
Trees get taller by the extension of growth at the branch tips. The trunk and branches will get thicker, but they do not move. If you have a tree with a branch 4 feet from the ground, say, the center line of that branch will forever be 4 feet from the ground. In time it will become thicker, but it does not move.
The knot of branches you have now will rapidly turn into a big ugly knob. As each branch gets thicker they will fuse together making this knob bigger. You can, of course, prune the ends to steer their future growth into whatever directions you might wish, but this won't change where they now join the tree. So, if you do nothing, a big ugly knob with a spray of branches emerging from it will become the prominent feature of your tree.
You need to choose two or three branches you want to keep and selectively remove all the others. If you do this immediately, the tree will likely grow into the attractive tree you want and you won't be able to tell this ever happened. The longer you delay the more difficult it will be to accomplish and the less likely it becomes that the result will be pleasing.