Heathcock was drafted three times in the MLB Draft: January 1979 by the Milwaukee Brewers (2nd round), June 1979 by the San Diego Padres (first round, 14th overall), and the Houston Astros in June 1980 (the first pick of the first round). He had gone to Golden West College in Huntington Beach, California before spending a year at Oral Roberts University.
Heathcock first played in the minors with the Astros' single-A club in Daytona Beach in the Florida State League. He went 9-0 with a 1.27 ERA before being promoted to the Astros' AA club in Columbus with the Southern League. He went 4-7 with a 5.08 ERA. He stayed on the club the following year and went 13-13 with a 5.61 ERA, and he then went 4-4 with a 3.15 ERA in 1983 before being moved up to the Tucson Toros in the AAA Pacific Coast League, where he went 10-3 with a 2.77 ERA.
Heathcock appeared on and off in four seasons for the Astros, debuting at the age of 23. He pitched two innings against the Chicago Cubs as a call-up on September 3, 1983, allowing three hits and two runs with a strikeout. He appeared in six games (three as a starter), garnering a save and a 2-1 record with twelve strikeouts to four walks.

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