I was going to recommend that next step exactly. This falls into a grey area. If you were still caring for her (i.e. she was in your clinic) or if your care of her extended over a long period of time, then it's a solid no. But this is a patient you helped care for a few days, with supervision in an ICU. You have no ongoing clinical relationship with her. Hence, in my book, I think you could pursue this if you want although you do need to be careful, and getting your PD's input will be helpful.
I will add several important issues:
1. Starting immediately, no matter what you do / what happens, you can have NOTHING to do with her ongoing care unless it's an emergency and you're the only option. Even if you decide not to pursue this, you can't care for her anymore. You should not answer her medical questions nor should you try to "help" fix any problems that might occur (i.e. she needs an appointment, or having trouble getting a refill, etc).
2. You mentioned that she was admitted for something else. Hopefully, you did not log into the EMR and review her chart. If you did, this makes dating her riskier if things go badly.
When considering something like this, the best way to look at it is to assume the worst. Assume that you date her, things get super ugly, she dumps you. You want to make sure that your actions can't be interpreted as abusing your position. So any of the following would make me worry:
A. If as part of your care of her you needed a social or sexual history and now know something "secret" about her.
B. You performed a vaginal, rectal, or breast exam on her, or any other exam that would be considered invasive.
C. You reviewed psychiatric notes in her chart
Even without any of this, if this relationship becomes ugly it could blow back on you because there's always some privacy boundaries that physicians cross (appropriately) when caring for patients.
