jebr

These are two very different questions.

What was his responsibility? Assuming the general facts of the OP's post are true (being at the gate at T-15, and not a second later, and that the information boards and the board at the gate all showed the wrong gate having the flight,) there's nothing more that he has to do.

But, assuming that the OP wants to have the best chance of making his flight, he should have done a few things differently. First, if he's arriving at the gate at T-15, the gates as advertised were close-ish, and it only took a few minutes to get a cup of coffee, the connection time was likely only about 30 minutes. That's not enough time to make a pit stop when flying an airline he hasn't flown in years, which may have different practices than he's accustomed to. At a connection, if you want to have the best chance of making it you go to the gate first, make sure there's no unexpected changes, and then if time allows go and grab food/drink/whatever. Then you go back to the gate around the start of boarding (usually T-40ish) and wait for the boarding call.

Cutting it close means that there's the risk of even one more unexpected issue causing a missed flight, with the only clear remedy being taking the next flight with available space and dealing with the delay. There's nothing more required by law or by Delta's contract of carriage.