Does it look bad?

I have been wondering about that too. Maybe wait a few more weeks. Or check in with HR first to find out, and let the HR person know that you are still interested in the 1st job, but just applied to 2nd job as a 2nd choice FYI. Maybe they would appreciate you letting them know first.

No.

I applied to two jobs at the same hospital. I didn't get one job, I got the other one...this wasn't under the same recruiter.

I suggest casting your net wider in terms of other options outside of the hospital you applied to if you don't want to run into any complications.

It is common practice to apply for multiple jobs at the same time. Depending upon the application system, this may mean that you have to actually create a separate application for each job rather than just indicating multiple jobs (you are applying for) on a single application. If the directions are not clear on their employment site, just call HR and ask them how to do it.

I was informed by a nurse recruiter to go ahead and apply for other positions the following week if I didn't hear back from the position I originally applied for.

No, apply to everything that interests you. I've even had recruiters help with finding a position that would be a good fit for me, and they weren't even the recruiters for the position. Getting to know recruiters can be very helpful, especially in a tight market.