Tom Hardy and Russell Brand have also pulled their fingers out - four to be precise - for the #1in4 campaign.
Alison Kirkham, the BBC's controller of factual commissioning, said one in four were affected by mental health issues.
"I'm delighted so many high-profile people have come together to raise awareness of this important cause."
Radio 1's Nick Grimshaw and Frozen voice star Kristen Bell have also snapped their digits to show solidarity.
"I wanted to get involved with the #1in4 campaign to help people realise they're not alone," said Grimshaw.
"Having mental health issues is nothing to be ashamed about."
The initiative accompanies the BBC's Minds Matter season, a range of programmes across TV and radio about mental health.
They include Mind Over Marathon, a documentary featuring the Duke of Cambridge that will air later on BBC One.
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Mr Trump pointed to the fact that during the caucus the Cruz campaign told voters rival Ben Carson planned to quit the race, which was not true.
The Cruz campaign apologised to Carson's camp, saying it was a misunderstanding.
Mr Trump also cited Cruz leaflets that accused Iowans of "voting violations".
Paul Pate, the top election official in Iowa, condemned the mailers on Saturday, saying the state does not track or grade individual voters.
Mr Pate said Mr Cruz's mailers "misrepresent Iowa election law" and that they were "not in keeping in the spirit of the Iowa Caucuses," but he stopped short of any official action.
"Ted Cruz didn't win Iowa, he stole it. That is why all of the polls were so wrong and why he got far more votes than anticipated. Bad!" Mr Trump said.
Earlier, he wrote on Twitter that Mr Cruz "illegally" won the caucus, but later deleted the tweet.
The reaction contrasts with his concession speech on Monday night, which was seen as a humble departure from his usual bombastic style.
Mr Trump placed second in Monday's contest, which he called "a long-shot great finish" in an earlier tweet.
Mr Cruz's camp is not taking the accusations too seriously.
"Reality just hit the reality star - he lost Iowa and now nobody is talking about him, so he's popping off on Twitter," Mr Cruz's communications director Rick Tyler told Politico in an email.
"There are support groups for Twitter addiction, perhaps he should find his local chapter."
There is no precedent for re-doing a caucus.
The remaining candidates are now in New Hampshire, the next state to hold a primary vote, where Mr Trump is leading in the polls.
Rand Paul, a Republican who represents the Libertarian wing on the party, dropped out of the race on Wednesday after finishing fifth in Iowa.

The National Portrait Gallery commissions feature people from the worlds of culture, politics, sport and fashion, including Dame Helen Mirren, Sir Paul Smith and Dame Kelly Holmes.
The works were all commissioned as part of the BP Portrait Award first prize.
'Paul Smith to J.K. Rowling' opens at the University of Hull on Wednesday.
More on this story and others from East Yorkshire and northern Lincolnshire
First prize winners of the BP Portrait Award are invited to create a portrait of a public figure who has made an important contribution to British history and culture.
The National Portrait Gallery said it was a "rare opportunity" to see all the commissions in one place.
Professor Glenn Burgess, acting vice-chancellor of the University of Hull, said: "Not only does the exhibition depict a wonderful range of famous faces, from actors to athletes and beyond, but the artists responsible have used an amazing variety of styles and compositions in creating these beautiful works."
The free exhibition runs at the Brynmor Jones Library until June.
Broadchurch stars David Tennant and Olivia Colman have posted selfies of themselves to show their support for a BBC mental health initiative.
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Presidential candidate Donald Trump has called for a new election in Iowa, accusing the Republican winner, Ted Cruz, of fraud.
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A collection of portraits depicting a host of household names is set to open in Hull as part of its city of culture celebrations.