Accies moved above Motherwell but stayed second-bottom in the Scottish Premiership following their fifth clean sheet in nine matches.
It was a sixth goalless draw of the campaign for Micky Mellon's United side, who stay sixth in the table.
Chances were at a premium in a competitive first half but each team had one golden opportunity.
United's came when a Dillon Powers cross found Harkes unmarked six yards out but the American midfielder's header was diverted over the bar by a diving Fulton.
Ross Callachan set up Callum Smith at the other end with a through ball which put the striker beyond his marker on the edge of the box. Smith took a touch but then fired over.
David Moyo shot well over from eight yards after chesting the ball down at the other end.
United striker Lawrence Shankland came close to adding to his spectacular midweek goal when he shot just wide with a 25-yard drive.
United stepped up the pressure as the match progressed but Hakeem Odoffin put in a couple of excellent penalty-box blocks as Accies stood firm.