The 33-year-old Spaniard crashed into a wall 20 laps into testing at Barcelona's Circuit de Catalunya yesterday 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.
"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.