Cameron resigned after Britain's narrow vote in the referendum of June 23 to leave the EU.
May became Britain's second female prime minister in history, after Margaret Thatcher.
Mr Davis said Article 50 should not be triggered immediately in case EU countries act "irrationally" and refuse tariff-free access to the European single market because the United Kingdom will not compromise on ending free movement so it can control immigration.
Russian President Vladimir Putin said he was ready for "constructive dialogue" with Britain's new premier, while White House spokesman Josh Earnest said USA officials who had worked with May "found her to be quite effective".
May vowed to tackle injustice, saying: "The government I lead will be driven not by the interests of the privileged few", and adding that "together we will build a better Britain". During the speech, he thanked his family and political advisers for their support over the years.
"When it comes to opportunity we won't entrench the advantages of the fortunate few".
Now she is set to name the new ministers in charge of the important health, education, Scotland, work and business briefs.
Mr Gove led the "Leave" side in Britain's European Union referendum battle alongside former London mayor, then betrayed him by making a bid for Conservative leadership - a job Mr Johnson had long sought.
The governing Conservative Party has coalesced behind May's leadership bid, bringing clarity at the national level, but concerns still remain, partly because the Brexit decision will impact the lives of millions in ways that are completely impossible to define at the moment.
In her first appointment after taking office, May named former foreign secretary Philip Hammond as Britain's finance minister.
In his final Prime Minister's Questions at Parliament, outgoing Prime Minister David Cameron opened Wednesday with the quip that "apart from one meeting in the afternoon with the Queen, my diary is remarkably light".
Outside Downing Street, a group of demonstrators chanted: 'What do we want?
Russian President Vladimir Putin yesterday thanked former British Prime Minister David Cameron in a valedictory message and congratulated his successor Mrs. Theresa May.
"It has not been an easy journey, and of course we have not got every decision right, but I do believe that today our country is much stronger".
Minutes later, the palace released a photo of May curtseying to the monarch and confirmed that the queen had "requested her to form a new administration".
The shock vote partly reflected discontent with European Union rules on freedom of movement that have contributed to record-high immigration - an issue on which May, as interior minister for the past six years, is politically vulnerable.
Boris Johnson has been appointed Foreign Secretary, Downing Street confirmed.
She is under pressure - both from pro-Brexit Conservatives and other European Union leaders - to start formal exit talks with the bloc.
"This whole process should be completed to allow triggering of Article 50 before or by the start of next year". So what's her recipe going to be for a divided Britain as she takes on as PM from David Cameron?
Trying to reclaim his legacy from his Brexit miscalculation, Cameron said his government had cut the deficit, overseen economic growth and legalized same-sex marriage.
Cameron told The Daily Telegraph it had been "a privilege to serve the country I love". "I will miss the barbs from the opposition".
Financial markets, which had been extremely volatile since the referendum, reacted positively to news on Monday that Ms May would become prime minister earlier than expected.