Can I simply add a loop of pex from my water heater, under my flooring, and back into itself? Would I need some sort of pump? I haven't laid the tile yet and water heater is right on the outside wall of the bathroom I'm building. Would I need to put this between the backer and the tile?
Adding hydronic radiant floor heating is a lot more complicated than you could imagine. To do a system for a bathroom running off of a domestic water tank, in operation for potable water requires all potable water components. Potable water components are expensive.
The storage tank isn't being used for domestic water. It's only being used for floor heating.
Other things that you need to consider are legionnaires bacteria radiant floor temperatures are right at the bacteria's ideal growth temperature of around 115 degrees Fahrenheit.
The Solution Electric radiant floor heating is your best bet.
This requires a little bit of electrical but is much simpler to install. It's also much thinner then pex.
A loop unto itself cannot flow.
Seriously. Imagine a hula-hoop full of water. Not much floor will get heated from that.
You need a suitable pump and thermostatic control, plus it may not be legal to tie heating into your potable water supply. Most floor and radiator heating systems are closed-loop and contain anti-freeze and anti-microbial treatments.
I tried heating my garage in the winter using my water heater and all copper piping connected to finned tube radiation. What a waste of my time and money. With the water heater temperature set at 130 degrees and a small bronze pump the finned tube radiation yielded almost no heating. If you want to heat the floor it can be done but you will need a lot of under the floor tubing and not just a few feet of it due to the relatively low water tank temperature. If you have a hot air furnace you can heat the basement fairly well if you do it correctly