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.
I have been working on a diode resistor gate that I call Light Logic and with a single gate I can create all eight basic gates, Buffer, NOT, AND, NAND, OR, NOR, XOR and XNOR. My project is posted on Hackaday as, [SHEDDING A BIT OF LIGHT ON SOME LOGIC](https://hackaday.com/?s=light%20logic). Not fast but it proves that DRL can do it all if folks do not restrict themselves to signal diodes and resistors. Think out of the box. Very basically a Light Logic gate is a LED coupled to a photo resistor/LDR. This combination acts as a switch much as a NPN transistor. Input 1N914 diodes are wired ahead of the LED and power and output are wired to the LDR much as a DTL gate. Granted the LDR does have a pronounced reaction time but this is a new way to create gates and my goal is a 100 percent transistor and relay free processor. Point, keep stray light from being exposed to the LDR.