Pickford ripped Ings's top and left him with damage to his body when attempting to clear a ball outside his box in the second half of Saturday's match.
Gallagher does not agree with Hughes's assertion, though, telling Sky Sports News: "I don't think this is a foul. The player gets the ball quite clearly. The goalkeeper goes to win the ball, he gets the ball and he's in the air, and his boot catches the shirt.
"I don't think any referee is going to give a foul for that. The goalkeeper has to go for the ball and he's cleared the ball. He's got tangled up in the shirt, and the referee was right to play on."
Gallagher also suggested that Theo Walcott's second goal of the afternoon was wrongly ruled out for offside.