Play was scheduled to begin at 12:30 BST but was regularly delayed because of intermittent showers over London.
Frenchman Gasquet lost 7-6 (7-2) 6-2 to American Steve Johnson in a stop-start match that finished after 19:30 BST.
British number four Dan Evans was due to face France's Paul-Henri Mathieu but they will now play on Tuesday.
n Monday's other completed matches, South Africa's Kevin Anderson and Canada's Vasek Pospisil won second-round qualifiers to reach the main draw.
Top seed Andy Murray faces Frenchman Nicolas Mahut - the world number 49 - in his first-round match, which is scheduled for Tuesday.
Top seed Andy Murray faces Frenchman Nicolas Mahut - the world number 49 - in his first-round match, which is scheduled for Tuesday.
