In the episode we see Picard tear off a sign, 'Romulans Only' and stomp on it, to challenge their tribalism. In that sense he picks the fight. The Romulans talk of their prior knowledge of Picard. They drag him outside and put a sword in his hand.
This leaves two possibilities:
Sword fighting is a common way to solve matters of honor in Romulan culture.
Is there a precedent for Romulans settling honor battles in a sword fight?