Does the obligation of Friday prayer fall away for a resident doctor if its time conflicts with a mandatory meeting at the hospital?

If your attendance at the Friday prayer does not entail harm to yourself or your livelihood, then you have no excuse for not attending it. You must repent and commit to attending it at its prescribed time (the beginning of the Dhuhr prayer time), or attend the second Friday prayer if you cannot make the first. However, if you are unable to attend either of the two Friday prayers, and you are able to change your specialization or workplace to be able to perform the obligation without harm, then you must do so.