By Dave Graham and Adriana Barrera MEXICO CITY (Reuters) – Mexico's Congress on Thursday overwhelmingly voted to open up the country's oil and gas sector to private investment in the biggest overhaul of the industry since it was nationalized in 1938. After a whirlwind final passage through Congress, President Enrique Pena Nieto's bill will offer companies the chance to operate oil wells, commercialize crude and partner with state oil giant Pemex as Mexico seeks to revive flagging output. It aims to entice oil majors such as Exxon Mobil Corp and BP Plc with production- and profit-sharing, service contracts and licenses. Mexico's two biggest parties faced down accusations they were betraying their homeland to foreign oil firms, and approved a series of changes to the constitution that could radically transform the fortunes of the world's No. 10 oil producer.