He said he takes "full responsibility" for the scandal but sought to play down his personal involvement in the gatherings detailed in the report by the senior official.
The Gray report gave details of gatherings at which officials drank so much they were sick, sang karaoke, became involved in altercations and abused security and cleaning staff at a time when millions of people across the country were unable to see friends and family.
Mr Johnson told a Downing Street press conference: "I understand why people are indignant and why people have been angry at what took place."
But when pressed on whether he ever considered resigning, he responded: "I overwhelmingly feel it is my job to get on and deliver.
"No matter how bitter and painful that the conclusions of this may be - and they are - and no matter how humbling they are, I have got to keep moving forward and the Government has got to keep moving. And we are."
Tory MPs gave a muted response to the report but a snap poll from YouGov after it was published said three in five Britons want Mr Johnson to quit.
One damning new detail was the "multiple examples of a lack of respect and poor treatment" of cleaning and security staff during the events, which Ms Gray said was "unacceptable".
Mr Johnson apologised, described their treatment as "repugnant" and "utterly intolerable" and said he has started "to make some inquiries" to find out who was behind the behaviour.
The report said the "senior leadership" in No 10 must "bear responsibility" for the culture which led to lockdown rules being broken at a series of events in 2020 and 2021.
The Prime Minister faced fresh demands to resign after the report said the public would be "dismayed" by the behaviour uncovered.
The Metropolitan Police has issued 126 fines for rule breaches in No 10 and Whitehall, with the Prime Minister receiving a single fixed-penalty notice for his birthday party in the Cabinet Room in June 2020.
But senior civil servant Ms Gray condemned the wider culture that had been allowed to develop under Mr Johnson's leadership.
She said some of the more junior officials who attended parties "believed that their involvement in some of these events was permitted given the attendance of senior leaders".
"The senior leadership at the centre, both political and official, must bear responsibility for this culture," Ms Gray said.
"Many will be dismayed that behaviour of this kind took place on this scale at the heart of government.
"The public have a right to expect the very highest standards of behaviour in such places and clearly what happened fell well short of this."
Labour leader Sir Keir Starmer told the Commons the report "laid bare the rot" in No 10 and called on Tory MPs to tell Mr Johnson "the game is up" and that it is "time to pack his bags".
In a Commons statement, Mr Johnson repeated his apology over the birthday party and added: "I take full responsibility for everything that took place on my watch.
"Sue Gray's report has emphasised that it is up to the political leadership in Number 10 to take ultimate responsibility and, of course, I do."
Mr Johnson said he was "humbled" by the experience and had learned his lesson.