Parents of the first four boys freed on Sunday have been able to visit them but had to wear protective suits and stand 2 metres (7 feet) away as a precaution.
Thongchai said one from the last group rescued on Tuesday had a lung infection and they were all given vaccinations for rabies and tetanus.
Prime Minister Prayuth Chan-ocha asked that the boys be given time and space to recover.
"The important thing is ... personal space," Prayuth told reporters.
"The best way is not to bother them and let them study."
The group ventured into the vast cave complex in the northern province of Chiang Rai after soccer practice on June 23 and were trapped when a rainy season downpour flooded tunnels.
They were lost for nine days before they were discovered by British divers on July 2.
Getting them out - which involved teaching boys as young as 11 who were not strong swimmers to dive through narrow, submerged passages - proved a monumental challenge. A former member of Thailand's navy SEAL unit died during a mission in the cave on Friday.
Rescue mission chief Narongsak Osottanakorn told a news conference the boys were just being children when they got lost and no one was to blame.
"We don't see the children as at fault or as heroes. They are children being children, it was an accident," he said.
He said falling oxygen levels inside the cave complex had added a sense of urgency to the rescue.
The commander of the navy SEAL unit that oversaw the rescue, Rear Admiral Apakorn Yuukongkaew, hailed the international effort.
"We are not heroes. This mission was successful because of cooperation from everyone," he said. "For SEALs, this is what we were trained for. The navy has a motto: 'We don't abandon the people'.”
Official help came from Britain, the United States, Japan, Laos, Myanmar, China and Australia, a government document showed. There were volunteers from Denmark, Germany, Belgium, Canada, Ukraine and Finland.
A video of the boys in hospital was show at the news conference. Some, wearing surgical masks, lay on their beds. Some sat.
The rescue has dominated front-page headlines in Thailand and beyond for days.
A senior Australian police officer acknowledged the degree of international cooperation "in a very unfriendly environment".
"It is amazing what the human being can do. There are extraordinary people doing extraordinary things," Glenn McEwan, the Australian Federal Police's Asia manager, told reporters in Chiang Rai.
"We are humbled to have been a part of it. Returning the Wild Boar soccer team safely into the arms of their loved ones is the good news of the year," he said.
"Hooyah! Mission accomplished," read one headline, echoing the rallying cry of the SEAL unit.
The hashtag #Hooyah was hugely popular on social media with people showing their support for the hundreds of rescuers, including divers from around the world, who helped to get the boys out.
The fate of the boys has even resonated as far as Russia, where soccer's World Cup is reaching its final stages. Players from France and England welcomed news of the rescue and sent their best wishes to the "Wild Boars" on Twitter.
"This victory goes to the heroes of the day, well done boys, you are so strong," French midfielder Paul Pogba tweeted after his team beat Belgium 1-0 on Tuesday to reach the final.
Manchester City and England defender Kyle Walker, whose team faces Croatia in the second semi-final later on Wednesday, said he wanted to send shirts to the boys.
"Amazing news that all of the Thai kids are out of the cave safely!" Walker tweeted.
A Google search on Tuesday for the words "Thai cave rescue" revealed 359 million results.
Araya Hargate, one of Thailand's top actresses, posted a cartoon of the boys surrounded by rescuers on her Instagram page, which has 7.9 million followers.
"After all ... the world is not such a bad place #humanityfaithrestored #thailandcaverescue," the actress wrote.
(Reporting by Panu Wongcha-um, Patpicha Tanakasempipat, James Pomfret and John Geddie in CHIANG RAI Additional reporting by Amy Sawitta Lefevre, Aukkarapon Niyomyat, Panarat Thepgumpanat and Chayut Setboonsarng in BANGKOK, and Colin Packham in SYDNEY Writing by Amy Sawitta Lefevre and John Geddie Editing by Paul Tait, Robert Birsel) ((firstname.lastname@example.org; Tel: +66 26489737; Reuters Messaging: email@example.com))