The episode, which has kindled controversy in the Italian media and arguments between centre-left and centre-right politicians, occurred on Sunday in Venice's Ca' Rezzonico museum, which houses 18th-century Venetian art.
"I'm sorry for what happened and if she ever wants to return to our museum, she will be more than welcome," director Filippo Pedrocco told Reuters by telephone from Venice.
The woman, visiting the museum with her husband and children, had cleared security when she entered the building.
When she reached the second floor, a room guard told her she must remove her "niqab," which leaves only the eyes visible, or leave.
"The room guard was over-zealous. He should not have done it. She already passed security and his only duty was to guarantee the safety of the artwork in his room," Pedrocco said.
The woman was believed to have been a member of a well-off family visiting Venice from Saudi Arabia or a Gulf state.
She refused to take off the veil and left the building, which faces Venice's Grand Canal and houses works by such 18th century Venetian masters as Giandomenico Tiepolo.
Source: Reuters (English)