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: 
It's certainly possible to make a computer without transistors, using diode logic for most functions. All computers prior to [1953](http://en.wikipedia.org/wiki/Manchester_computers#Transistor_Computer) avoided transistors, and some of these heavily used diode logic.

But eventually you need some form of amplification and inversion.

Inversion you can easily achieve using transformers (at least, if you are passing discrete pulses rather than continuous logic levels through the logic. This was common in the 1940s and 50s) - simply swap the secondary winding connections.

Amplification : assuming you have ruled out valves (vacuum tubes) as well as transistors, you are limited in your options. Relays are an obvious choice, for clock rates up to a few Hz. Above that, there are tricks you can play on transformers to amplify current changes by using smaller currents in other windings to bring their cores in and out of saturation. I've never heard of anyone exploiting this form of "magnetic amplifier" for computing, so it may not be possible. 

On the other hand, the [Elliot 803](http://en.wikipedia.org/wiki/Elliot_803) was a transistor computer, but it implemented its logic functions using magnetic cores, with just one transistor per gate to provide gain.
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...?