"It was disappointing for the crowd - they made a lot of noise," Butcher told BBC Scotland.
"That is good for us but we didn't do the job for them on the pitch, which is something we will have to work on."
Inverness's best efforts were a half chance for Nick Ross and an Adam Rooney attempt that Andy Graham cleared as Hamilton put on a solid performance to take their first three points of the season.
There's always a danger that when a team like Hamilton have had two heavy defeats and have lost in the cup, that they'll give the response the manager is looking for Inverness manager Terry Butcher Accies crashed out of the Co-operative Insurance on Tuesday after a 1-0 defeat by Raith Rovers, and Butcher believes it contributed to their performance at the Caledonian Stadium.
"There's always a danger that when a team like Hamilton have had two heavy defeats and have lost in the cup, that they'll give the response the manager is looking for," said Butcher.
"We didn't get the response we were looking for on the pitch.
"We were excellent against Celtic and even better against Dundee United. It's a lesson you sometimes have to take in the SPL.
"We didn't get our game going after the goal, it was Hamilton who got the goal and hung on well and defended the lead.
"We didn't have that luxury. We were always chasing the game."