Question: Is it possible to build a computer (Turing complete) using only diode logic without transistors? I know DTL was a thing, but from what I could tell, they used transistors to amplify signals.

Answer: 
Impossible. With nothing but diodes, and I suppose you allow resistors, the output levels of any chunk of logic will span a smaller range than the input levels. Forward voltage drops would add up until there'd be no signal. There has to be amplification in every gate, or at least in many places.

The biggest show stopper though, is that with just diodes there'd be no way to invert a signal. That means no XOR gates, or half-adders and full-adders, no way to test if two bits are the same or different. You'd have to design a diode circuit where if the input goes up, the output goes down, and by at least as much as the input went up.

Finally, there'd be no way to store a bit. There has to be some way to maintain state, such as a program counter, registers, call stacks or something equivalent. Flip-flops are easy to make with cross-connected NOR or NAND gates. But we don't have those in pure diode logic. 

That said, it doesn't mean a little bit of diode logic isn't helpful. A couple diodes can make a cheap little OR gate in a TTL circuit, if done right, saving a chip that might be only 1/4 utilized. (In fact, I had a two diode OR gate in my science fair project, years ago.)

Now, since getting larger voltages and inversion of signals is important, I'm starting to wonder - if you allow inductors, you can invert voltages and create voltages outside the range of the inputs. Although still passive components, thus losing energy every step of the way, I wonder if there might be some fun to be had contemplating diode-inductor logic...?
This is a tough question. I know that "AND" gates can be made from diodes and that single pull-double throw relays can provide the inversion and amplification. So it looks like it is possible (theoretically)! However, it should be noted that diode logic cannot act as a direct replacement for normal transistor logic in most scenarios due the fact that it uses a path to ground instead of the input being pulled high (or low, as with a PNP transistor). Anyways, good luck!