Embedded is a LARGE field, and ranges from really little stuff to things that are really just a desktop machine in a funny box.
I'm going to go with 3 categories of embedded systems, to correspond with my experience, then I'll make some recommendations.
Small embedded systems are things run by micro-controller type devices. They don't have any OS, scheduler, etc, etc. You're running code running directly on the CPU, and in general you'll have main loop that gets started at power on and never exits. There are a lot of options for micro-controllers, ranging from things like the venerable 8051 to the really tiny PIC up and a lot of products based on the ARM cores. These are nice because you have complete and direct control over every facet of the system's operation. Of course, that also means you need to be careful about things like timing and latency of operations, and how long you can spend on any one thing before you go back to your main loop.
Medium embedded systems are bigger things, and which will have an embedded OS such as Wind River's pSOS (there are MANY choices for this, I happen to have used pSOS once upon a time). I'm calling it medium because at this point you're NOT running a single main loop, and actually have a scheduler. The hardware may not be much different from the higher end of the previous group, but the presence of a scheduler and a more formalized driver architecture really improves your ability as a programmer to get stuff done. On the downside, it might be more trouble to meet really tight timing because you can easily introduce more unexpected delays from the scheduler. Also, you have to be careful that your hardware does things that make sense for the OS you are using - for instance, not all interrupt strategies work with all embedded OSs. At the larger end of this group I'll put in Windows CE, the embedded offering from Microsoft. This segment is getting squeezed out to some degree as more and more processing power is becoming available for less and less money.
Big embedded systems are even bigger and run an OS that a desktop user might well recognize. Linux is common here, as is the embedded (or even the regular!) version of MS Windows. There's a lot of this going around in the medical field (look at a lot of ultrasound machines). The advantage here is that you can run a full bore installation of Linux, with all the useful stuff that a real OS brings on very small and very cheap hardware anymore.
Now, being your first foray into embedded, and given that it's medical (where I suspect you have margins to support reasonable hardware budgets), I recommend that you use some small computer like a rasberry pi, a beaglebone, or something else based on one of the multitude of ARM cores out there. Run Linux on it, and build your app on top of that. In fact, you probably want to simply leave your device as-is, package it into a box with the small computer and communicate via USB. Effectively, just port your existing app to Linux and go.