In his own lifetime, Galileo's advocacy for the Copernican view was controversial. He repeatedly ran afoul of the Roman Inquisition, until he was finally tried in 1633. He was forced to recant, and placed under house arrest.
The arrival of Grantville, and the spread of up-time ideas and knowledge, caused Galileo Galilei's heresy trial to take a different track. At the start of 1634, it had not yet gotten started, although Pope Urban VIII had convened a Commission of Inquiry consisting of Church astronomers and Inquisition cardinals.
In 1634, up-time Americans became involved in two separate ways. Pope Urban appointed Lawrence Mazzare, then the United States of Europe's ambassador to the Most Serene Republic of Venice, to defend Galileo. At the same time, Frank, Ron, and Gerry Stone became involved in the Venice Committee of Correspondence's plan to rescue Galileo.
- Galileo Galilei at the Harry Turtledove Wiki