continued to receive visitors until 1642, when, after suffering fever and heart palpitations, he died on January 8, 1642, at age 77. The Grand Duke of Tuscany, Ferdinando II, wished to bury him in the main body of the Basilica of Santa Croce, next to the tombs of his father and other ancestors, and to erect a marble mausoleum in his honour. These plans were scrapped, however, after Pope Urban VIII and his nephew, Cardinal Francesco Barberini, protested, because Galileo was condemned by the Catholic Church for "vehement suspicion of heresy". He was instead buried in a small room next to the novices' chapel at the end of a corridor from the southern transept of the basilica to the sacristy. He was reburied in the main body of the basilica in 1737 after a monument had been erected there in his honour; during this move, three fingers and a tooth were removed from his remains. One of these fingers, the middle finger from Galileo's right hand, is currently on exhibition at the Museo Galileo in Florence, Italy.
Source: Wikipedia
There seems to be a lot of curiosity on who is the woman that is buried nice to Galileo. No one knows, but there is thought that it is his eldest daughter Virginia which is a nun. There is no proof that this is true. It is said that it would cost a lot of money to find out the truth. I am sure one day we will all find out.