Maybe basing a life's work off of a hastily scrawled compliment someone wrote on a cocktail napkin after your high school play is a bit childish…but that's exactly why Riggan decided to dedicate his life to acting.
Carver's napkin-compliment didn't just inspire Riggan's adaptation of a Carver story; it was the genesis of his career as an actor in the first place. So, needless to say, the napkin has a lot of significance for Riggan, which is why Mike disrespecting it hurts so much:
MIKE: It's on a cocktail napkin. He was f***ing drunk, man.
Oof. Riggan's a bit stunned that a compliment by the great Raymond Carver—who he's now paying homage to—could simply be disregarded as a drunken joke.
So what does he do with the napkin? He tries to show it to Tabitha to start off a nice conversation about the play. After all, it's a great anecdote that most columnists would love to use to introduce a piece.
But like Mike, Tabitha want's nothing to do with it; she doesn't even look at it. And after Tabitha leaves without drinking her martini, Riggan decides to finish it for her…and slams the empty martini glass down right on Carver's napkin before he leaves.
The napkin's gone full circle. It started supporting the condensation of Carver's alcoholic beverage, became an inspiration for a young boy, a cherished memorabilia of an aging actor, and finally, a cocktail napkin once more.
This happens right before the deepest part of Riggan's depression and self-doubt. We can only guess how having something that had given him purpose and direction and affirmation all his life totally disregarded played a role in this decline.