Franco's liner, which had an exit velocity of 100 mph, struck Gausman and ricocheted to catcher Gabriel Moreno, who retired Franco at first to end the inning, leaving the bases loaded.
The team announced that Gausman had a right ankle contusion and X-rays came back negative. His status for his next start is uncertain.
"We'll see how he feels tomorrow and go from there," Blue Jays manager Charlie Montoyo said.
Gausman stumbled forward and fell face down in front of the mound after being struck. He received attention from the trainer before getting to his feet and slowly walking off the field.
He signed a five-year, $110 million contract with the Blue Jays last offseason. He came into the start 6-6 with a 2.93 ERA.
Gausman allowed four hits in two scoreless innings before departing. He walked one and struck out three.
Rays left-hander Shane McClanahan, who beat the Blue Jays in Game 1, offered an unprompted expression of concern for Gausman at the end of his postgame media session.
"I saw Gausman's X-rays were negative," McClanahan said. "That's a great thing to see. I was asking about him all through the game, how he was doing. That's the best of a bad situation. We all care for that guy and we're praying and hoping for the best."
Right-hander Casey Lawrence, who joined the Blue Jays as the 27th man for the doubleheader, replaced Gausman in the third.