Spain's conservative ruling Popular Party (PP) has called for an end to criticism of King Juan Carlos who broke a hip on a hunting trip to Botswana.
The king, 74, was flown back to Spain for surgery after falling on a step in the early hours of Friday morning.
A leading Socialist, Tomas Gomez, said he should choose between "public responsibilities or an abdication".
But PP Secretary General Maria Dolores de Cospedal warned against "casting doubt" on the monarchy.
"To use one particular event to cast doubt on institutions does no favours to Spanish democracy or to our form of state and government," she said.
Although the royal family did not confirm the reason for his trip to Botswana, the country is well known as a destination for elephant-hunting, leaving the monarch open to criticism on two levels.
Firstly, his trip came at a time when Spaniards were facing harsh economic austerity, raising questions about the monarchy's lifestyle.
Secondly, Juan Carlos is honorary president of the Spanish branch of conservation group WWF.
Spanish newspapers have published pictures dating back to 2006 in which the king is seen standing with a gun next to a dead elephant.
An online petition on the actuable.es website said more than 46,000 people had backed a petition calling for the king's resignation from WWF.
'Duty of service'
Mr Gomez, head of the Socialists in Madrid, told reporters that the king had to "choose between his obligations and the duty of service of his public responsibilities, or an abdication that would allow him to enjoy a different kind of life".
However the national leader of the Socialist party, Alfredo Perez Rubalcaba, refused to comment on the matter.
"We don't comment on the king's private life, whether it seems good or bad to us," he said.
Queen Sofia, Princess Letizia and Prime Minister Mariano Rajoy were among visitors to San Jose University hospital in Madrid on Monday, where the king's condition was said to have quickly improved after Saturday's hip replacement operation.
After returning from an Easter holiday in Greece, the queen spent 25 minutes in the hospital and said that doctors had described his progress as "phenomenal".
Hospital director Javier de Joz told reporters that his mobility was "practically complete and natural".
It is not the first time that King Juan Carlos's taste for hunting has landed him in trouble.
In 2006, an official in the Vologda region in north-eastern Russia alleged that he had shot dead a tame bear that had been plied with vodka, although royal officials said the claim was ridiculous.
Last week, the king's grandson Felipe Juan Froilan, 13, had to be treated in hospital after shooting himself in the foot while doing target practice outside a family home at the end of his Easter holiday.