The Accies manager called on his team to be brave in the meeting with Steven Gerrard’s unbeaten side.
At times, Hamilton played three v three at the back and were eventually rewarded for their courage with a stoppage-time goal from Ross Callachan after being thwarted on numerous occasions by Allan McGregor in the Rangers goal.
“I think that was the least we deserved out of the match,” Rice told Sky Sports. “No disrespect to Rangers. I asked my lads to be brave, I asked them to be on the front foot, go and have a go. Don’t have any regrets.
"Between now and the end of the season, go and have a real go.
"I think you saw a team today playing for each other, a team that believed they could get something out of the game and a team that deserved a goal in the last minute.
“You know you’ve got to take your chances in these games. We didn’t manage manage to do that, Allan had some really good saves.
"Rangers get the goal you think it is going to be one of those days. Even if we had been beat today I would have been proud of what they put in today. I couldn’t ask for any more.
"I’m just delighted that Ross popped up with the goal.”
Hamilton impresses for Accies
The result took Accies to within three points of 11th-place Ross County with two games in hand. The first of which comes during the week with a trip to Livingston, starting a run of four away games.
There were a number of star performers for Accies in the draw, with Bruce Anderson being a constant pest in attack but there was special praise for teenage centre-back Jamie Hamilton.
“I believe in these lads,” Rice said.
"I think you saw in young Jamie Hamilton today. Absolutely outstanding for an 18-year-old, absolutely fantastic. He needs the help of the experienced lads. It’s trying to get the blend right. I think we are nearly there.
“Jamie has certainly improved with Aaron Martin alongside him, Lee Hodson, talking to him. “