French Open Men's Tennis
Jun 2 French Championships Men's Tennis: In an all-Australian final Rod Laver beats Roy Emerson 3-6, 2-6, 6-3, 9-7, 6-2; 2nd leg of Laver's 1st Grand Slam
- Jun 2 French Championships Women's Tennis: In an all-Australian final Margaret Smith beats doubles partner Lesley Turner 6-3, 3-6, 7-5
Belmont Stakes
Jun 9 94th Belmont: Bill Shoemaker aboard Jaipur wins in 2:28.8
- Jun 17 FIFA World Cup Final, Estadio Nacional, Santiago, Chile: Brazil go 1-down early but recover to beat Czechoslovakia, 3-1
US Golf Open
Jun 17 US Open Men's Golf, Oakmont CC: Jack Nicklaus wins his first major title by 3 strokes in an 18-hole playoff with Arnold Palmer
- Jun 24 Outfielder Jack Reed hits his only MLB home run in the 22nd-inning as NY Yankees edge Detroit Tigers, 9-7 in slowest extra-inning game in league history; 7:00 hours
- Jun 26 Boston Red Sox Earl Wilson no-hits LA Angels, 2-0
- Jun 27 Dutch soccer clubs Rapid JC and Roda Sports merge to form Roda JC Kerkrade; KNVB Cup winners 1996–97, 1999–2000
- Jun 29 Frank Howard hits the 5,000th Dodger home run
Sports History
Jun 30 LA Dodgers' future Baseball Hall of Fame pitcher Sandy Koufax no-hits New York Mets, 5-0; first of 4 career no-hitters thrown by Koufax
- Jun 30 US Open Women's Golf, Dunes Golf & Beach Club: Murle Lindstrom wins her first LPGA Tour victory, 2 strokes ahead of runners-up Ruth Jessen and Jo Ann Prentice