"The day Suthep turns himself in to police is the day we the red shirts go home," Nattawut, a red-shirt protest leader, told cheering crowds on Monday.
Another protest leader, Jatuporn Prompan, said Vejjajiva, who has parliamentary immunity, should also be prosecuted.
The announcement came after the Vejjajiva had pleaded over the weekend for an end to the street protests that have paralysed parts of the capital, and for an acceptance of his reconciliation plan that offers new elections a year ahead of schedule.
Nattawut began his announcement by saying protesters "unconditionally accept" Abhisit's offer to dissolve parliament in late September ahead of November elections.
He then demanded that Suthep face charges for the April 10 clash.
Abhisit has warned he will scrap the plan for early elections if the protesters do not leave their vast base, which has been fortified with barricades made from piles of fuel-soaked tyres, bamboo poles and razor wire.
The rivals are under pressure from the international community to find a peaceful way out of the tense standoff which has left the country reeling from the worst political violence in almost two decades.
In the latest incident, two police officers were killed over the weekend in gun and grenade attacks, while a bank and the home of the Thai election commission chief were targeted with a grenade and small bombs on Sunday.
Police said nobody was hurt in the attacks, and the reds shirts denied they were involved.
The reds shirts consider Abhisit's administration undemocratic because it came to power in a 2008 parliamentary vote after a controversial court ruling ousted elected allies of Thaksin Shinwatra, who was himself unseated in a 2006 coup.
They have said the government is intent on clinging to power until at least September to ensure the new army leadership line-up is appointed and the national budget is approved in parliament.