The former prime minister criticised her successor for threatening to override Britain’s treaty obligations in the Brexit divorce settlement and his failure to commit to targets for defence spending and international aid.
Ms May drew comparisons with the murder of PC Keith Palmer in a terrorist attack outside the Houses of Parliament in March 2017.
“What happened in Washington was not the act of a lone extremist or a secretive cell, but an assault by a partisan mob whipped up by an elected president,” she wrote.
“I know from experience that leaving power is not easy – especially when you feel that there is more you want to do.
“But anyone who has the honour of serving in such a position must always remember that the office is bigger than the individual.
“The peaceful transfer of power is a hallmark of any democracy; it is what makes us special.”
Ms May said the arrival of Mr Biden in the White House represented a “golden opportunity” for Britain, with the return of a more normal style of presidency.
However, for the full potential of “Global Britain” to be realised in this new era there needed to be a change in international affairs and an end to the “absolutism” which said “if you are not 100 per cent for me then you must be 100 per cent against me”, she wrote.
She added: “In this world there is no room for mature compromise. Indeed, compromise is seen as a dirty word. In fact, the opposite is true.
“Strong leadership knows when to compromise to achieve a greater good. If the world is to work together to ‘build back better’ then we must all be willing to compromise.
“We must reject a scene in which a few strongmen face off against each other and instead bring people together in a common cause.”
Ms May said that while the UK was well placed to play a decisive role in shaping a more cooperative world, she suggested it had not been helped by Mr Johnson’s actions.
She again strongly criticised his threat – later withdrawn – to override elements relating to Northern Ireland in the Brexit Withdrawal Agreement and his decision to drop the commitment to spend 0.7 per cent of national income on international aid.
“To lead we must live up to our values,” she said. “Threatening to break international law by going back on a treaty we had just signed and abandoning our position of global moral leadership as the only major economy to meet both the 2 per cent defence spending target and the 0.7 per cent international aid target were not actions which, in my view, raised our credibility in the eyes of the world.
“Other countries listen to what we say not simply because of who we are, but because of what we do. The world does not owe us a prominent place on its stage.
“Whatever the rhetoric we deploy, it is our actions which count. So, we should do nothing which signals a retreat from our global commitments.”
Downing Street said it rejected Ms May’s criticism that the UK had abandoned its position of “global moral leadership”.
The prime minister’s official spokesman said: “I would reject that characterisation.
“The UK is and will continue to be an outward-looking nation and will continue to be a leading voice on the international stage.
“I would point to the work we have undertaken on climate change, both home and abroad, where we were one of the first countries to commit to net zero by 2050.
“You’ve seen the role that we’ve played in response to the pandemic and the action we’ve taken to support Covax (a global initiative to ensure vaccines reach those in greatest need) and other issues where we’ve played a big part on the international stage.”
The No 10 spokesman said he was not aware of any conversations the prime minister had held with Ms May since taking office.