No. The Medici became the royal family of Florence and Tuscany in the 16th century, having started as merchants and merchant bankers. They held the title of Grand Dukes.
However, the male line of the Medici family died out in the 18th century, but there are still female-line descendants in existence. The name is common enough, but if you meet someone named Medici and he tries to tell you he's a direct descendant of the famed Medici family, he's either lying or just misinformed.