Gregory was the youngest of three children born in Kansas City, Missouri. His father died when Gregory was three years old, and he left school before completing his senior year. At nineteen he married Luella Simpson. After inheriting a substantial fortune, he purchased a Mercury-powered Allard and made his racing debut in November 1952 at Caddo Mills, Texas, retiring with head gasket failure. Undeterred, he switched to a Chrysler hemi-powered car for Sebring in 1953, and his first win came in just his third race at Stillwater, Oklahoma. Driving a Jaguar, he accumulated further victories in America, including the Guardsmans Trophy at Golden Gate Park, San Francisco, and a race at Offutt Air Force Base in Omaha, Nebraska.
By the end of 1953, Gregory was invited to international competition, finishing 14th at the 1954 1000 km Buenos Aires. He spent 1954 and 1955 racing Ferraris in Europe, claiming the inaugural Nassau Trophy at the Bahamas Speed Week in 1954. Returning to America in 1956, he won numerous SCCA events, and in 1957 triumphed at the Argentine 1000 km, which earned him a seat with Scuderia Centro Sud in Formula One, driving the Maserati 250F.
Gregory's Formula One debut came at the 1957 Monaco Grand Prix, where he scored a third-place finish — the first podium for an American in an F1 Grand Prix. He followed that with fourth at both the Pescara and Italian Grands Prix, ending 1957 sixth in the championship despite contesting only half the rounds. A move to Cooper-Climax in 1959 alongside Jack Brabham and Bruce McLaren brought two podiums: third in the Dutch Grand Prix and a career-best second at the Portuguese Grand Prix. That year, teammate Brabham claimed the World Championship and Cooper won the Constructors' title. Gregory also took pole position and set a course record at the non-championship Aintree race.
His early years were marked by serious crashes, frequently the result of pushing unreliable machinery beyond its limits. He flipped a Maserati at the 1957 Venezuelan Grand Prix, and in 1959 broke his leg and shoulder jumping clear of a doomed car, which cost him the final two races of the season. He also famously exited cars mid-crash by leaping free at the last moment, a habit that gave rise to his flamboyant reputation. After 1960, however, his style matured and he became known as an elegant and careful driver.
Gregory continued in Formula One until 1965, largely with uncompetitive privateer teams. His best championship result in the latter period was sixth at the 1962 United States Grand Prix at Watkins Glen driving a Lotus 24 for UDT Laystall. He also won the non-championship 1962 Kanonloppet race at Karlskoga, Sweden. In total he started 43 World Championship Grands Prix.
In parallel with his Formula One campaign, Gregory was one of the most active sportscar competitors of his era. He entered 16 editions of the 24 Hours of Le Mans between 1955 and 1972, setting the overall fastest lap at the 1960 race and finishing fifth in 1961 in a Porsche RS61 Spyder. In 1961 he won the 1000 km Nürburgring alongside Lloyd "Lucky" Casner in a Maserati Tipo 61 for the Camoradi Racing Team. In 1962, he won the Canadian Grand Prix sports car race at Mosport Park in a Lotus 19-Climax.
The pinnacle of Gregory's sportscar career came in 1965, when he partnered with Austrian Jochen Rindt — who would become the 1970 Formula One World Champion — to win the 24 Hours of Le Mans in a North American Racing Team Ferrari 250 LM. That same year Gregory also raced at the Indianapolis 500, starting from the back of the grid and working his way to fifth position before a mechanical retirement.
Gregory gradually wound down his racing activities after 1965, recording a notable second place at the 1966 1000 km Monza alongside John Whitmore. Following the death of his close friend Jo Bonnier at the 1972 Le Mans race, Gregory retired from competition. He moved to Amsterdam, where he worked as a diamond merchant before later operating a glassware business. He died on November 8, 1985, of a heart attack at his winter home in Porto Ercole, Italy.
Gregory is one of only nineteen drivers in motorsport history to have attempted all three legs of the Triple Crown — the Indianapolis 500, the 24 Hours of Le Mans, and the Monaco Grand Prix — having won at least one of those events. He was inducted into the Missouri Sports Hall of Fame in 2005, the Kansas City C.A.R.B. Hall of Fame in 2007, the Watkins Glen Walk of Fame in 2012, and the Motorsports Hall of Fame of America in 2013.