The gunman fired five shots at close range and Mr Fico was hit in the stomach and in the arm.
He was rushed to hospital in an air ambulance and spent five hours in surgery, conducted by surgical and trauma teams, according to Ms Lapunikova.
Later on Wednesday, Mr Taraba told the BBC's Newshour programme that Mr Fico was "not in [a] life-threatening situation at this moment".
He added the prime minister was shot "from very close" and that "one bullet went through the stomach and the second one hit the joint".
How Robert Fico rose to dominate Slovak politics
Police have not yet identified the alleged suspect. Unconfirmed local media reports say he was a 71-year-old writer and political activist.
A video being widely circulated on Slovak media purports to feature the suspect.
In the footage, the man says he disagrees with government policy and its stance towards state media. The BBC does not know if the person in the video is the perpetrator who was detained at the scene nor the circumstances under which it was filmed.
The shooting came on the day parliament began discussing the government's proposal to abolish Slovakia's public broadcaster RTVS.
Thousands of Slovaks have protested against the proposed reform of the public broadcaster in recent weeks. However, a planned opposition-led demonstration was called off on Wednesday as news of the shooting emerged.