TEHRAN — President Hassan Rouhani of Iran stood his ground on Wednesday in an escalating regional showdown, defending a Yemeni rebel missile attack on the Saudi capital that Saudi Arabia has denounced as an Iranian “act of war.”

Saudi forces were “constantly bombing” Yemen, Mr. Rouhani said, adding: “What reaction can the nation of Yemen show toward this amount of bombardment? They say that they should not use weapons? Well, you stop the bombs, and then see if you don’t get a positive reaction from the nation of Yemen.”

The conflict in Yemen, described by the United Nations as the world’s worst humanitarian crisis, is one of several proxy battles between the two powers, with Iran supporting Houthi rebels who have controlled much of Yemen since 2014 and Saudi Arabia fighting to restore the government of President Abdu Rabbu Mansour Hadi.

Over recent days, however, aggressive moves in Yemen and from Saudi Arabia’s crown prince, Mohammed bin Salman, seem to have pushed the cold war for regional dominance into a new and dangerous stage.