I've read this whole thread, and I'm surprised about Dad's hasty decision to cancel the appt. with the specialist. It was a VERY good idea to go to the diving cardiologist WITH him. You were on the right track, for sure. But, I suspect CONTROL is a big issue here. He just may not want you to "hear" what the Dr. might have said.
In the last couple of years I have had to deal with aging parents-in-law. I found that they really needed "an advocate" to go into the Dr.'s appts. with them. They both failed to report significant symptoms, they "downplayed" their symptoms, they didn't understand exactly what the Dr. was saying, forgot stuff, "simplified" things by what they actually "thought" they heard, and reported to family that everything was OK. They split pills in half to save money. MIL took half a pill every other day, then TOLD the DR. that's what she's doing, and reported back that the Dr. said that was OK. Well, it wasn't "OK", but the Dr. couldn't convince her that what she was doing was not OK, so she continued to do exactly as she pleased....not what the Dr. ordered.
I find that as people age, and they are very unsophisticated with modern medicine, they just simply don't "hear" the importance or reasoning behind treatments. It may very well be that the cardiologist has recommended some tests or treatments that will be very effective in making your Dad better, but he may be "refusing" them, because he doesn't fully understand the benefits, or the risks of ignoring them. And, just for the record, and "older" husband and wife going together to the Dr. is not much better than either of them going alone.
Bottom line.....I FIRMLY believe that all people need someone to be an "active" participant in their health care, go to their appts. with them, know what meds they are on, know why they are taking them, and follow through to see that they take them in the proper dosage, on the proper schedule, etc. They can "remind" them about things the Dr. told them, but they never "heard. "Older people" very much includes people who are of full mental strength, but may be "stubborn" or very ill-informed about current medical issues. And, always, always, always have a family member present at all times a loved one is in the hospital. *stepping off my soap box, now*