He was chosen 5th overall by the Washington Capitals in the 1980 NHL Entry Draft. He played six seasons for the Capitals, four of them split with the Capitals' farm team Hershey Bears. Veitch never became a star with the Capitals. A trade to the Detroit Red Wings in 1985 gave him a new lease on life. In 1986-87 he scored 13 goals and 45 assists (an NHL career high).But the management blamed him (and 5 others) for the Wings' elimination from the playoffs. Veitch was traded to the Toronto Maple Leafs. He scored 48 goals and 209 assists in 511 NHL games played.