He played for Michigan three years under coach Bud Middaugh, from 1985 to 1988, leading them to two Big Ten championships. In 1987, he won the James E. Sullivan Award as the top amateur athlete in the United States, becoming the first baseball player to win the award. Abbott was the flag-bearer for the United States at the 1987 Pan American Games in Indianapolis, helping lead the USA to a second-place finish. Baseball was a demonstration sport in the 1988 Summer Olympics; Abbott pitched the final game, winning an unofficial gold medal for the United States. Abbott was voted the Big Ten Athlete of the Year in 1988. He would be selected 8th overall by the California Angels in the 1988 draft. Abbott's University of Michigan #31 jersey was retired at the Wolverines' April 18, 2009 home game against Michigan State University. 
In 1991, Abbott won 18 games with the Angels while posting an ERA of 2.89, finishing third in the American League Cy Young Award voting. In the 1992 season, he posted a 2.77 ERA, but his win-loss record fell to 7-15 for the sixth-place Angels. Abbott also won the Tony Conigliaro Award in 1992.
He struggled through the 1996 season, posting a 2–18 record with a 7.48 ERA and briefly retired.
Abbott returned to the White Sox in 1998, starting five games and winning all five. Abbott continued his comeback the following year with the Milwaukee Brewers, but pitched ineffectively. This was the first time he'd played for a National League team, forcing him to bat for the first time in his career. He recorded two hits in 21 at-bats during his Brewers stint.
Abbott retired after the 1999 season with a career record of 87–108, with a 4.25 ERA.
When preparing to pitch the ball, Abbott would rest his mitt on the end of his right forearm. After releasing the ball, he would quickly slip his hand into the mitt, usually in time to field any balls that a two-handed pitcher would be able to field. Then he would secure the mitt between his right forearm and torso, slip his hand out of the mitt, and remove the ball from the mitt, usually in time to throw out the runner at first or sometimes even start a double play. At all levels, teams tried to exploit his fielding disadvantage by repeatedly bunting to him; this tactic was never effective.
1992 - Abbott was awarded the Tony Conigliaro Award, given annually by the Boston Red Sox to a Major League player who overcomes an obstacle and adversity through the attributes of spirit, determination, and courage that were trademarks of the Boston star.