His comments come as the health secretary, Matt Hancock, told the House of Commons the Government expects the outbreak to "get worse before it gets better", adding "dealing with this disease is a marathon, not a sprint." He also announced a round of new funding for the NHS to help deal with the spread of the virus.
On his way back to the UK, he stopped off for several days at a French ski chalet, where five Britons were subsequently infected with the virus.
There are eight confirmed cases of the virus in the UK so far.
How the virus spread: movements of "super-spreader" from Singapore to central London hospital
Speaking from hospital, Mr Walsh said in a statement: "I would like to thank the NHS for their help and care - whilst I have fully recovered, my thoughts are with others who have contracted coronavirus.
"As soon as I knew I had been exposed to a confirmed case of coronavirus, I contacted my GP, NHS 111 and Public Health England."
"I was advised to attend an isolated room at hospital, despite showing no symptoms, and subsequently self-isolated at home as instructed," he continued.
"When the diagnosis was confirmed I was sent to an isolation unit in hospital, where I remain, and, as a precaution, my family was also asked to isolate themselves.
"I also thank friends, family and colleagues for their support during recent weeks and I ask the media to respect our privacy."
The gas analytics company Mr Walsh works for, Servomex, released a statement, in which it said: "We are very pleased that Steve Walsh has made a full recovery. We continue to provide support to him and his family.
We are working with Public Health authorities to ensure the welfare of our staff and communities and wish anyone with the virus a quick and full recovery."
A spokeswoman for the Scout Association said: "We are aware that Steve Walsh, from the Brighton area, who volunteers with the Scout movement, is suffering from coronavirus.
"This volunteer has not been to any Scout meetings since his return to the UK.
"We wish Steve well and hope he recovers soon."
The comments came as virus was given an official name of Covid-19, by the director general of the World Health Organisation, Tedros Adhanom Ghebreyesus.
Dr Ghebreyesus told a press conference a name was decided that "did not refer to a geographical location, an animal, an individual or group of people, and which is also pronounceable and related to the disease".
He said: "Having a name matters to prevent the use of other names that can be inaccurate or stigmatising.
"It also gives us a standard format to use for any future coronavirus outbreaks."