This is a tough one; I don’t envy your position here. Often when a person refuses a move to an assisted living it’s based on a strong desire to remain
in his/her home and maintain the lifestyle he/she has grown accustomed to, mixed in with some fear of the unknown. Understand that you can’t get anywhere until you acknowledge your father’s position and begin to talk with him about how the thought of this transition makes him feel.
Use this time in which your mother is hospitalized to talk with your father (based on experience, you should have at least a few days and even longer if she were to go to rehab upon discharge). Forget trying to convince him that assisted living is best and focus instead on hearing him out and acknowledging the concerns that you can imagine you’d have yourself if you were in his shoes. Demonstrating an understanding of his position first, should open up the communication further. When it does, share your concerns for your mom and what you know to be her reasons for wanting to go to an assisted living now. Don't argue them, just lay them out there and allow him time to respond. If you can, tell him that you don't know anything about assisted living facilities and ask if he would take a tour with you so that you can both learn more. Do you best to assure him that there are no strings attached and that you fully appreciate this is his decision and not yours. If he agrees to the tour, move as fast as you can to make this happen and try to see at least two for a comparison.
If he won’t take a tour, or if after a tour he is still adamantly refusing, understand this might not be the time. You will have planted a seed though and that is no small feat. If you (they) can afford it you might also try talking with your mother about going to an assisted living alone, just on a trial basis, while your father remains in their home. If your parents have a close relationship, the separation may be a motivating factor for him to join her.
If your mother will not consider leaving him even for a trial, encourage them to hire enough help so that your mother’s growing needs are adequately met and that they are safe. Then wait for another opportunity to explore a transition to assisted living. There are no magic bullets here and these conversations are very difficult. What I've suggested is really the best anyone could do given the situation.
All the best to you.