I've found that it almost never works to ask a cancer patient "Are you depressed?" It's difficult to suggest to your father that he's having trouble coping without him personalizing
it (thinking, "Gosh, I'm just not trying hard enough," for example). After all, nobody wants to be a cancer patient, and certainly nobody wants to be an instant psychiatric patient on top of that.
What I find works much better is to ask other questions that get at the way depression affects peoples' lives. That way you're talking about what's happening, rather than making another scary diagnosis. You might ask your father, "How are you sleeping?" "How's your appetite?" "Are you feeling worried a lot?" If he mentions memory problems, you might follow up on that by asking, "How's your concentration these days?" Once you get him talking about the specific issues, you might say, "You know, there are medications that can help you sleep better and worry less and have more appetite. Let's talk to the doctor about them."
If your father just seems in generally low spirits, you might try asking him, "How are your mood and spirits holding up through all this?" I find that many patients are incredibly relieved to be asked this question because it gives them a chance to talk in an open-ended way. Often older patients will answer by saying, "I don't know what's wrong with me, but I lose my temper all the time," or "I cry at every little thing."
I also find it's reassuring to patients to focus on the chemical abnormalities that underlie depression, because then it seems more like any other illness -- which is how it should be viewed anyway. So I'll say, "Have you heard of a medication called Lexapro? It can increase the serotonin levels in the brain that have gotten low during treatment. Once we replace the serotonin in your bloodstream, that extra grumpiness and inability to concentrate will likely go away."
When I'm talking to patients, I don't even necessarily use the word antidepressant. But I will let patients know that sometimes these medications are used to treat depression, and I'll talk to them about side effects and alternatives. I find it's not as important to give their depression a name as it is to identify symptoms and offer treatment and hope. There are many new antidepressants prescribed for sleep, memory, and other conditions, so it really doesn't matter if we prescribe them for a diagnosis of major depression or for managing the symptoms that often accompany depression.