The political future of Andrej Babiš, the scandal-tainted Czech prime minister, has been plunged into fresh doubt after his son said he had been lured to Crimea and abducted to stop him testifying about alleged criminal fraud in his father’s business dealings.

Opposition parties demanded a parliamentary no-confidence vote on Babiš’s minority coalition government after the Czech news site Seznam Zprávy broadcast a documentary that included footage of his son, Andrej Babiš Jr, making dramatic allegations apparently implicating his father.

Babiš, 35, tracked down to Switzerland, where he was living with his mother, the prime minister’s first wife, said he had been persuaded to go to Crimea at a time when police were seeking to question him. Babiš, who has a history of mental health problems, had been given the choice of “taking an extended holiday” or being admitted to a psychiatric hospital.

Once in Crimea he was forcibly detained by two Russians, one of whom was a psychiatrist who had previously treated him during a stay in a Prague mental hospital, he claimed.

Babiš has been charged, along with his billionaire father, in the long-running investigation into allegations that nearly €2m (£1.7m) in EU funds was falsely obtained by one of the latter’s businesses, Čapí Hnízdo (Stork’s Nest), a hotel and conference centre south of Prague.

However, police have still to interrogate the prime minister’s son, having been unsure of his whereabouts after he left the Czech Republic, leaving the prosecution effectively stalled.

The money was awarded as a small business grant after Čapí Hnízdo was transferred from Babiš Sr’s sprawling Agrofert company to undisclosed new ownership in 2007. It later emerged that the new owners were the prime minister’s son and daughter, along with his partner of the time and future wife, Monika, and her brother. The complex was subsequently repossessed by Agrofert, a conglomerate of 230 companies including chemical, food, energy and media firms.

In an email exchange with two Czech journalists who spent a year tracking him down, Babiš Jr admitted to being “technically” connected to Čapí Hnízdo. He also said he had been made to sign some papers on the matter but did not understand what they were.

Amid a political outcry, the supreme state attorney’s office said it would investigate the son’s claims.

The prime minister, attending an international conference on Libya, in the Sicilian city of Palermo, issued an angry denial, calling the report “a manipulation”, an “outrageous” attempt by journalists to pressure law enforcement authorities, and “part of a script to destroy me and get me out of politics”.

Writing on Facebook he said his son was mentally ill and on medication and also that his daughter, Andrea, had bi-polar disorder. “No one kidnapped my son, he left the Czech Republic voluntarily,” he wrote, adding that police had investigated the alleged kidnapping and dismissed it.

The Czech prime minister, the country’s second richest man, worth an estimated US$3.5bn (£2.7bn) heads a minority coalition consisting of his Action of Dissatisfied Citizens (ANO) party and the Social Democrats (ČSSD), which relies on the Communist party (KSCM) to stay in power. Other parties have refused to serve in a Babiš-led government because of the charges facing him.

The fate of a no-confidence motion would almost certainly depend on the Social Democrats, the coalition’s junior partner. The party said it would speak to Babiš first before deciding.

Jiří Pehe, a political analyst at New York University, in Prague, said Babiš would be weakened even if he survived a confidence vote. “Before, he could always say the whole thing was a plot and he’d done nothing wrong,” said Pehe. “Now there’s a video that goes to the core of something emotional. Even if Czechs don’t believe the son, they see that there is something very wrong in the prime minister’s family.”