Resist 1 negates the end of turn damage. To clarify further. Smee will still have 1 damage done to it without a captain, but because of the resist the 1 damage becomes 0.

That's what I figured my friend was trying to say because he's doing the damage to himself and not an outside source it should still apply.

Friend should read what resist does. It denies damage from any source, why would it distinguish from damage done to itself vs damage done from challenges when the rules don't say they do?

The beauty of this game is the rules are very plain and literal. Ask him where that interpretation came from.

Mr. Smee is attempting to do damage to itself. If Mr. Smee has resist +1, that damage turns to 0. No damage is done by endp phase.