How can I increase the level of my blue shirts?

There's actually a pretty simple way to ensure your medical team is constantly healing someone. You've been on the right track with the grenade, but you didn't take it far enough.

What it requires is the cooperation of several other members of your crew; they can all be medical, it doesn't matter. Get your hapless red shirt to throw his grenade. After you self-damage your ship, punish him by making him stand in the resulting fire. Now, crew are smart enough to move out of fire if possible, so what you need to do is surround the spot with three crew members, move him in, and then close the box with the last crew member. Viola! Instant red shirt barbeque, with unlimited potential for healing.

You will need to have an enginner or two around to keep the fire in check, though, to make sure the rest of your staff don't turn out extra crispy. That's easy to do, though, just make sure not to put out the fire your red shirt is so helpfully baking himself in.