The 33-year-old Spaniard crashed into a wall 20 laps into testing at Barcelona's Circuit de Catalunya on Sunday and was airlifted to hospital under sedation.
Alonso suffered concussion but the team said scans on the 2005 and 2006 world champion were "completely normal."
Alonso would be kept in hospital, McLaren added without giving details of his injuries nor how long he would remain there.
His manager Luis Garcia Abad put up a picture of a smiling Alonso sitting up in bed on Twitter.
Hora de comer! Muchas gracias por vuestro apoyo! Lunch time! Thank you for your support! pic.twitter.com/v6cUCfKXNL— Luis Garcia Abad (@lsgrcbd) February 23, 2015
"He will remain here the time we need to be sure everything is right," the manager told reporters at the hospital.
"The impact was quite hard. We have to be sure everything is fine, so I can't say if it is one, two or three days more," Garcia Abad added.
The Twitter photo was accompanied by the message: "Lunch time! Thank you for your support!"
"Fernando Alonso is making a solid recovery in hospital, and is chatting to family, friends and hospital staff," said a McLaren-Honda team statement.
"A thorough and complete analysis of his condition was performed, involving CT scans and MRI scans, all of which were completely normal," the team said.
"In order to provide the privacy and tranquillity required to facilitate a peaceful recuperation, he is being kept in hospital for further observation, and to recover from the effects of the medication that successfully managed his routine sedation yesterday.
"We intend to give him every opportunity to make a rapid and complete recovery, and will evaluate in due course whether or not he will participate in the next Barcelona test."
The team strongly denied what it called "the erroneous rumours that have spread recently to the effect that Fernando was rendered unconscious by an electrical fault."
McLaren-Honda blamed strong winds on the track for the accident.
The car initially hit trouble after it went onto artificial turf that lines the edge of the track.
"A consequent loss of traction caused a degree of instability, spitting it back towards the inside of the circuit, where it regained traction and struck the wall side-on," said the statement.
"Our findings indicate that the accident was caused by the unpredictably gusty winds at that part of the circuit at that time, and which had affected other drivers," said the statement.
"We can categorically state that there is no evidence that indicates that Fernando's car suffered mechanical failure of any kind," it added.
Cars can reach speeds of up to 150 miles (240 kilometres) per hour on the third bend at the track where Alonso crashed. Other drivers said he was going slower however.
Alonso ended a five-year stay at Ferrari to join McLaren for the 2015 season.
He is linking up with another former world champion Jenson Button as the team renews its partnership with Honda in a bid to get back to returning to something like its old form after two seasons in the doldrums - their last win came in the season ending 2012 Brazil Grand Prix.