MIAMI — Senator Marco Rubio of Florida, a charismatic Republican whose name has circulated recently as a possible vice-presidential candidate, lashed out on Friday at those who accused him of embellishing his family’s history.

Responding to his critics, he asserted that his family’s inability to return to Cuba was a defining event in all their lives, regardless of when his parents first arrived in the United States.

“The pain of my parents’ permanent separation from the nation of their birth, their inability to visit there and move there, was a major part of our upbringing,” Senator Rubio said in an interview. “They were immigrants, and they were also exiles. That is the essence of my story.”

But Senator Rubio acknowledged that he was wrong about the date, an inaccuracy he attributed to the fact that his parents always spoke generally about when they first arrived “in the 1950s.” His working-class parents came to the United States during a tumultuous time in Cuba and hoped to improve their lot here, always with an eye toward going back home, he said.