Chants of "Levein must go" and "Craig Levein, we want you to go" were heard around Tynecastle Park towards the end of the match. Hearts were ahead twice through Sean Clare and Christophe Berra but defended poorly at both George Oakley's equalisers for Hamilton.
The visitors could easily have won the match and struck the crossbar through Marios Ogkmpoe. Hearts fans took their anger out on Levein after a result which left them joint-bottom of the league with two points from four games - and only four Premiership wins in 2019.
“I understand that. There is a sense of frustration with the supporters and I get that. I'm not going to criticise them for voicing their opinion, but I believe I can fix this and get us up the table pretty quickly," said Levein.
"To play here is not easy. Every time we sign a player we talk about the stadium. When things are going really well it's fantastic. When things aren't going well, it's very difficult to rise above and shine when other people are feeling a bit nervous."
Levein does not plan any more signing before the transfer window closes on Monday night and will proceed with the squad he has.
"I don't think we need that," he added. "We need to get a few bodies back from injury but once we do I think we have a fairly strong squad. That's not an excuse in any way shape or form but I think we have players more than capable of climbing the league table."
The Hamilton manager Brian Rice felt his team should have won the match after his imposing strike pairing of Oakley and Ogkmpoe troubled Hearts. “Anybody in the stadium would say we could have won that game with the chances we had,” said Rice.
“I'm delighted to get a point but I'm a little disappointed we didn't get three. Maybe that's me wearing my Hamilton hat. I thought the performance merited three points.
“I like to play two up front. It causes problems and it's a really difficult shift for one guy on his own. I have two lads up there who are big and physical and strong. The two of them can run.
“We saw George can finish and Marios had plenty chances. It gives the rest of the lads confidence when the ball goes up there that the forwards can keep it.”