I would tell her why you tried to stop being her friend. But that assumes you want to be friends again. If somehow you cannot be friends because you still feel jealously.

But I didn't agree with you in the first place. Both the not being friends anymore and the way you decided to try to stop being friends.

And now she wished you happy birthday, well happy birthday to you, I am kind of confused how you wished you would have completely ignored her. If you know for sure someone is trying to be friends with you, why not let them know if you don't want to. Even if you want to hide your personal reasons, just say so. Why try to completely ignore someone who was a friend and tries to be a friend?

So I do not know why you are still at war with yourself. Why aren't you relieved? I don't know exactly what you send her or exactly what you think you want. But why not tell her everything? If she is offended by the truth, you get what you want. You don't need to be her friend anymore. If she understands and appreciates your honestly, you have a friend again and you are both on the same page without any hard feelings or things to worry about. Except of course you cannot do something similar every few months. You say 'ignoring her for so long' but it was only a few weeks or a month? I recently reconnected with someone I didn't see for 12 years.

And as for the source of your jealousy, if you want to be successful professionally, you better surround yourself with people who are already successful. If all your friends are losers then yes you will be the most successful one among your friends, but they may drag you down. If you associate with people who already achieved what you want to achieve, just associating with them will help you in indirect ways. Their mindset will rub off on you. Yes, jealously isn't the best emotion. But it is not like you are jealous of her boyfriend. It is not like it is completely impossible for you to both work at the same prestigious hospital.