There are of course many ways of saying "therefore" in Greek, all of which have to do with causality: "such and such is the case, which therefore makes so and so exist." In the sentence you bring forward, however, causality does not play a role. The act of diagnosis does not cause being: instead, it is an activity through which we recognize our existence (as human beings). In other words, diagnosis is a sign through which we know that we exist as diagnosing entities. So οὖν etc. won't work here. Latin is much looser: so Descartes could say cogito ergo sum and mean "I think, and thus draw the conclusion that I am." Ergo can play that game for him. Ancient Greek can't do that with οὖν etc.
BTW the ancient Greeks might have understood what Descartes was getting at, but probably wouldn't be able to make any sense of "we diagnose therefore we are." For them, diagnostics was a given, and being was a given, but it wouldn't have occurred to them to interrelate the two.