Rajoy recalled the King's role in helping to restore democracy following the death of dictator General Francisco Franco in 1975 and the decisive action he took to fend off an attempted military coup on February 23rd 1981.
On that day a section of the army seized parliament, firing shots over the heads of lawmakers, in a bid to establish another military regime.
Within hours the King — who is the head of state and commander-in-chief of the Spanish armed forces — appeared on live television in full military regalia and ordered the coup plotters back to their barracks.
"The role which the King had in Spain's transition to democracy, and after, during the undesirable events of February 23rd 1981, which we wish never took place, show who the monarch is. They demonstrate the strength of the institution," he said a news conference, which he had originally planned to hold jointly with British Prime Minister David Cameron.
Cameron, however, did not participate after hearing the news of the death of Lady Margaret Thatcher.
"I certainly think that a large majority of Spaniards support an institution that has been very useful and very helpful for our country."
The King's popularity has dropped in recent months, especially among young people born after the attempted coup attempt, according to a poll published Sunday by El Pais newspaper.
A slim majority of Spaniards, 53 percent, disapproved of the way the 75-year-old King is carrying out his role, against 42 percent who approved.
That gave him an approval-versus-disapproval rating of -11, compared with +21 in December, the first time he has received a negative rating.
Among those aged 18 to 34, the King's approval-versus-disapproval rating was even higher, coming in at -41.
The King's popularity has suffered due to a luxurious elephant-hunting trip he took in Botswana last year while Spain is struggling through a steep recession and a corruption investigation involving his son-in-law Iñaki Urdangarin.
The case, opened at the end of 2011, is centred on allegations of embezzlement and influence-peddling against Urdangarin, who is married to the King's youngest daughter, the Infanta Cristina, and his former business partner Diego Torres.
The pair are suspected of overbilling regional governments to stage sports and tourism events, and then siphoning off money to the non-profit Nóos Institute, which Urdangarin chaired from 2004 to 2006.
Cristina was officially named a suspect in the case last week and ordered to be questioned. It is the first time a direct relation of the King has been called to appear in a court of law on suspicion of wrongdoing.