Antoine-Curier's penalty on the stroke of half-time cancelled out Dani Sanchez's close-range opener at the Tulloch Caledonian Stadium.
Visiting substitute Marco Paixao hit the post in the second half and Russell Duncan, off the bench for Inverness, had his low shot cleared off the line as each side looked for a winner.
Inverness and Hamilton both went into the game after defeats at Ibrox against defending champions Rangers, but Hamilton's 4-0 loss last Saturday was more punishing than the Highlanders' 1-0 midweek defeat.
Billy Reid made five changes to his Hamilton starting line-up as they aimed to bounce back, with Martin Canning, Gary McDonald, David Buchanan, Simon Mensing and Nigel Hasselbaink starting, David Elebert, Grant Gillespie and Marco Paixao moving to the bench, and Michael Devlin and Lee Kilday dropping out of the squad.
Inverness manager Terry Butcher named an unchanged starting line-up, but influential Irish midfielder Jonny Hayes was included on the bench after missing a month of action due to an ankle injury.
Spaniard Dani Sanchez scored his second goal in two home games when he broke the deadlock after a quiet opening.
The attacker lashed the ball into an unguarded net in 18 minutes.
A shot by Nick Ross from just inside the penalty area was spilled by Tomas Cerny and Inverness top scorer Adam Rooney reacted the quickest, striking the ball back towards goal.
Cerny put it into the path of Sanchez who fired home.
Youngster Ross might have doubled the advantage in 25 minutes when he forced a low diving save from Cerny.
Foran squared the ball to the 19-year-old, whose low drive from the edge of the box was thwarted.
Moments later former Inverness midfielder Dougie Imrie failed to test home goalkeeper Ryan Esson after a fast counter-attack saw him burst forward, but his shot ballooned over.
Imrie did call Esson into action with eight minutes to go in the half when his near-post cross was touched away by the shot-stopper following good work from Hasselbaink.
With Hamilton coming into the ascendancy, Inverness' Ross Tokely felled Gary McDonald in the box with only seconds remaining in the half.
Antoine-Curier coolly scored the equaliser from the spot on the stroke of half-time.
Hamilton started the second period brightly with Andrew Graham firing wide moments after the restart.
The defender sent his effort just off target after connecting with a cross from Flavio Paixao.
Sanchez might have scored his second goal of the game just after the hour when he latched onto a poor defensive clearance but shot wide.
Butcher made two changes after 64 minutes to try to inject more life into side, including replacing Rooney with Shane Sutherland. The other swap saw Duncan take over from Eric Odhiambo.
Two minutes Reid responded by taking off Antoine-Curier and introducing Gillespie.
Hayes made his comeback when he replaced Sanchez with a quarter of an hour to go as Inverness hunted a winner.
His involvement seemed to add some verve to the hosts' attack with Stuart Duff powering forward to find Ross who steered his shot just wide.
Substitute Marco Paixao could have scored with his first touch seconds after coming on but his free header hit the post.
Moments later Duncan had his low shot cleared off the line from sliding defender Martin Canning after the midfielder was played through by Hayes.
Neither side managed to find a winner in the closing stages as the game petered out in uninspiring fashion, but Hamilton are now just a win away from going level with second-bottom Hibernian.