The Tyneside club confirmed on Saturday that Alan Pardew had been released from his contract as manager to take over at Crystal Palace.
However, fans that had long called for Pardew's departure were unable to celebrate with a win as Newcastle delivered a lifeless performance at the King Power Stadium.
Carver's men had a 32nd-minute effort from Remy Cabella controversially disallowed for offside, before Leonardo Ulloa settled the tie in Leicester's favour with a header seven minutes later.
The defeat marks the third straight year that Newcastle have exited the FA Cup in the third round, and Carver conceded that their performance was not acceptable.
"Newcastle's fans should expect better. I'm one of them. I've been there with them and I would be the same," he said in quotes reported by BBC Radio 5 Live.
"They travelled all this way having spent a lot of money over the festive period, and they're entitled to their opinions."
And the 49-year-old was keen to make sure his players felt the fury of the travelling support at the final whistle.
"The one thing as far as I was concerned is that I wasn't going to let our players leave the pitch before going over and thanking the fans," he added.
"And I was right there at the front, and so I should be because I can take it, I'm a big boy."
