I had a Heart transplant at Phoenix and yes, all the rooms I was in were private and ICU also. As for recovery for day type surgery no its an area with curtains between recovery rooms. the private rooms is mainly for in-patient hospital stays. Based on the surgery your Husband is having I would guess He would stay a few days at least. But if you have access to the patient portal already you can send messages to the doctor's staff who are handling your husbands' surgery.
Good Luck and your at one of the Best Hospitals I have ever been to and Ive been in several.