Mr Walsh contracted the virus in Singapore before flying to France for a skiing holiday on 24 Jan, where he passed it on to 11 more people.
"As soon as I knew I had been exposed to a confirmed case of coronavirus, I contacted my GP", the 53-year-old said.
"I would like to thank the NHS for their help and care and although I have fully recovered, my thoughts are with the other people who have contracted coronavirus", he said.
"Anyone with a cough, fever, or shortness of breath who attends hospital and has recently returned from one of the specified countries, will be advised to follow signs to NHS 111 pods and call for advice, igso they stay isolated from other patients and avoid causing unnecessary pressure in A&E".
Last night the baby's father, who did not wish to be named, said his son had "all the symptoms" associated with the virus and feared his four-year-old daughter may also have been infected.
Mr Walsh was also linked with five more cases of coronavirus in the United Kingdom, including two doctors.
Walsh, who says he has fully recovered from the virus, thanked Britain's National Health Service.
An A&E worker at Worthing Hospital has tested positive for coronavirus, according to reports.
Meanwhile, three inmates at Bullingdon prison, in Oxfordshire, are being tested.
He said his family have been asked to isolate themselves "as a precaution".
Steve Walsh was being treated at the specialist infectious diseases unit at Guy's and St Thomas' NHS Foundation Trust in London but has now been given the all-clear to return home.
A statement on the website for County Oak Medical Centre - operator of previously closed Warmdene and Carden surgeries - told patients that the facilities have been "extensively and rigorously cleaned throughout".
A sign is pictured on the door of the Deneway branch of the Warmdene Doctor's Surgery in Brighton.
Mr Walsh travelled to the French Alps before returning to London from Geneva on an EasyJet flight carrying more than 100 people.
Also on Tuesday, Health Secretary Matt Hancock told MPs in the Commons that new funding was being launched immediately "to support any urgent works the NHS needs for the coronavirus response, such as the creation of further isolation areas and other necessary facilities". The 31-year-old was reportedly brought back to the United Kingdom two weeks ago after being extradited from Thailand.
He said: "One further patient in England has tested positive for novel coronavirus, bringing the total number of cases in the United Kingdom to nine".
Earlier, Dr Ghebreyesus, director-general of the World Health Organisation, described the outbreak as "a very grave threat for the rest of the world".