The leaked video caused anger among thousands of families of those who lost loved ones in the pandemic, with one labelling the government's behaviour "despicable".
Meanwhile, on Wednesday evening, Mr Johnson also announced new restrictions for England as the Omicron variant spreads.
He said the country would move to "Plan B", which will see people asked to work from home where possible, and vaccine passports will be needed for entry into nightclubs and places with large crowds.
Earlier in the day, Mr Johnson started off Prime Minister's Questions in the House of Commons with the apology and told MPs he had been "repeatedly assured" that no party had taken place and no COVID-19 rules had been broken.
"I understand and share the anger up and down the country at seeing No 10 staff seeming to make light of lockdown measures, and I can understand how infuriating it must be to think that the people who have been setting the rules have not been following the rules," Mr Johnson said.
"But, I repeat, I have been repeatedly assured since these allegations emerged, that there was no party, and that no COVID rules were broken, and that is what I've been repeatedly assured."