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.
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!