Matias Vecino snatched a point for slumping Inter, whose streak without a victory extended to six matches in Serie A.
Roma hit the front just past the half-hour mark at San Siro, Stephan El Shaarawy pouncing on a Davide Santon error after the full-back badly misjudged a long kick from goalkeeper Alisson before Vecino equalised four minutes from the end.
After the result left Inter fourth and 11 points behind leaders Napoli, Spalletti defended Santon and the Nerazzurri.
"Santon doesn't have to apologise to anyone. It's easy when you're sitting on your comfy sofa, quite another when running for 90 minutes alongside El Shaarawy," Spalletti told Mediaset Premium.
"He's a good lad, he trains well and mistakes can happen, especially on a ball that came that far. Nobody else helped him on that either, so it's not as if he was alone. He has no reason to apologise.
"The real issue here is that we are Inter and we must have the courage to fight it out with sides like Roma, Lazio and Napoli for a top-four finish. We have to take responsibility and make things happen. We can't just wait for the train to go past and grab it. I won't see us timorous or apologetic.
"Samir Handanovic didn't make the saves that Alisson did, so we played well. We made some errors in the quick passing and knew that Roma would be aggressive in their pressing game.
"We practise these things in training, but if they don't come off in a match situation, that will inevitably damage self-esteem. We have to work on what we are and what we do best, not losing the ball cheaply."