It was the awkward non-handshake that dominated the first official meeting between two of the world's most powerful leaders.

But US President Donald Trump's press secretary had a simple explanation for why he rebuffed German Chancellor Angela Merkel's request for a handshake in the White House.

"I don't believe he heard the request," Trump's spokesman Sean Spicer told German weekly Der Spiegel.

But the Germans aren't buying it.

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The country's top-selling Bild newspaper said that was "improbable", saying that throughout the meeting, Trump did not once look Merkel in the eye.

The difficult encounter came as a new row erupted over environment at a G20 meeting of finance ministers in Germany, when Trump's administration defied the international community by refusing to renew a pledge on climate change.

The visit between the world leaders began cordially enough, with the pair shaking hands at the White House entrance.

But Merkel's suggestion of another handshake in the Oval Office went unheard or ignored by Mr Trump - an awkward moment in what are usually highly scripted occasions.

Trump was attacked as misogynist by many on social media in the wake of the handshake snub.

In his two months in the Oval Office, he has been pictured shaking the hands of Canadian Prime Minister Justin Trudeau, where he offered his hand, and Japanese Prime Minister Shinzo Abe, which lasted for 19 apparently painful seconds for the Japanese leader.

There was also the awkward pre-inauguration handshake with his predecessor, Barack Obama, in the same office.

Trump hit back at his critics via his usual method, Twitter.

Germany's defence minister Von der Leyen, a close Merkel ally, on Sunday rebuffed Mr Trump's Twitter claims about the European country's debts.

"There is no account where debts are registered with NATO," she said in a statement, adding that NATO spending should not be the only criteria used to measure Germany's military efforts.

But it was not just Merkel's meeting with the president that lit up social media.

The German Chancellor, who has a PhD in chemistry, also sat down with Ivanka Trump, who has taken on something of a hybrid First Lady / First Daughter role in her father's presidency.

The model and fashion designer met with Merkel at the White house in a round table discussion with business leaders about vocational training.

It's not the first time Ivanka has joined the discussion with a world leader either.

Only last time, when she met with Canadian Prime Minister Trudeau, it was she doing the sideways glancing.