West Ham put in a deeply concerning display at the Amex on Saturday to leave them teetering back on the brink of the relegation zone.
“It’s a cliche to say it was not good enough but you have to apologise to the fans because we have let them and the club down,” said Hammers captain Rice.
“As a player, I am hurting, all the lads are hurting. Last week we won 4-0, and then played really well in the FA Cup – we should have won – so to have this result is a real setback.
“But credit to Brighton because they played us off the pitch and that was not a performance that was acceptable at all.”
West Ham now face two big fixtures in a week, against Cypriot side AEK Larnaca in the Europa Conference League on Thursday and then at home to Aston Villa on Sunday.
Rice said: “Brighton was not good enough. It is going to hurt. We have let the fans down massively…but we will be onto the next.
“We just keep our heads down and I said to all the lads, we listen to ourselves and the people around us, and we stick together as a group and move on to the next one.
“We know it was not good enough today but let’s try and bounce back on Thursday and then we will have Villa on Sunday. I am sure our fans will be behind us in these next two fixtures.”
Moyes had to endure chants of “You don’t know what you’re doing” from sections of the away end at the Amex but he retains the support of the West Ham board for now.
Moyes said: “We let ourselves down. That’s one of the worst results and worst performances since I’ve been here. I totally understand the fans’ reaction. They have been watching a team that has been doing really well over the last few seasons so it’s hard for them to watch that.”