The 39-year-old will spend a further two days in hospital - with his partner Linda Morselli by his side - as he recovers from surgery on his jaw and teeth.
His Italian model girlfriend has reportedly been keeping relatives updated on his condition via WhatsApp.
Taking to social media, Alonso said: "Thanks for all your wishes, I'm OK and looking forward to getting 2021 underway."
A statement from his Alpine F1 team - formerly Renault - said: "Following his cycling accident Fernando Alonso was kept under observation in hospital in Switzerland.
"Medics discovered a fracture in his upper jaw and conducted a successful corrective operation.
"The attending medical team are satisfied with his progress. Fernando will remain under observation in hospital for a further 48 hours.
"Looking forward, after a few days of complete rest, he will be able to progressively resume training.
"We expect him to be fully operational to undertake preparation for the season."
https://www.thesun.co.uk/sport/14026...k-jaw-surgery/