NHS staff and members of the public were filmed gathering outside Russells Hall Hospital, in Dudley, on Thursday night.
The crowd formed as a live singer performed outside the hospital as part of the Clap for Carers initiative, which has seen people celebrate key workers at 8pm each Thursday.
One video of the celebration, which was deleted and then reposted by Sam Jones, was viewed more than 48,000 in 20 hours and led to numerous people raising concerns.
One member of the public said they had written a complaint to to the Dudley Group NHS Trust which runs the hospital.
The complainant acknowledged the "selfless efforts and hard work" of frontline workers but said they were left "appalled" at the scenes.
They added: "When patients are suffering and dying, when families are unable to be with loved ones, how can you think it is okay to arrange and engage in a large social gathering, and not respect social distancing that goes against the lockdown rules.
"This is not okay. Who thought this was a good idea? This needs investigating and the staff, patients and community deserve an apology."
Social distancing guidelines set out by the Government urge people to stay two metres apart and "avoid large and small gatherings in public spaces".
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Diane Wake, chief executive of the Dudley Group trust, said: "We are overwhelmed by the support being shown us by our community and understand how people want to demonstrate this support for their hospital.
"This is a great source of strength for our staff, however our main concern is that people stay safe, and the best way to help us is to stay at home.
"Our staff at all levels take a few minutes out of their week each Thursday to share the weekly #Clapforcarers campaign with their colleagues. This happens at shift handover time, when people are coming to work or leaving the site. It occurs at hospitals and trusts up and down the country.
"We are acutely aware of the huge impact this virus is having on our whole country and individuals in our communities, and our condolences go to those who have lost loved ones during this really stressful time. The loss of any patient is very keenly felt by all those involved in their care.
"We also have a duty to care for the mental health of our workforce and allow them a time to share in the feelings of goodwill being displayed across the country.
"Our message to everyone is to stay at home and stay safe."
A number of police officers also attended the event but Chief Superintendent Sally Bourner, Dudley Police Commander, said they were just passing by.
Ch Spt Bourner said: "Our officers are in support of the great work NHS staff and other key workers do every day and officers across the force have been taking part in the Clap for Carers initiative where possible and appropriate.
"Officers did not attend Russells Hall Hospital last night specifically to join the applause.
"They were patrolling around the hospital - which is a key location for police presence during this national health crisis - and took a moment to pause to show their appreciation.
"We take social distancing very seriously and are working hard across our communities to engage with people to explain and encourage adherence with the restrictions."
In response to the concerns, the Care Quality Commission urged people to follow social distancing to stop coronavirus spreading.
A CQC spokesman said: "We understand the vast majority of people are following social distancing measures to help protect the NHS and we would urge anybody not following the guidelines to do so to help stop the spread of Covid-19."
Downing Street has urged people to follow social distancing guidance after the country’s most senior police officer was filmed clapping for carers on a crowded Westminster Bridge.
Metropolitan Police Commissioner Dame Cressida Dick led dozens of officers in the weekly applause for key workers.
The Prime Minister’s official spokesman said: “We would ask that everyone takes responsibility and adheres to social distancing rules so that we can safely show our appreciation for those who are working so hard to fight coronavirus.”
Meanwhile, Barnsley Hospital asked people not to come to the site to clap NHS staff after crowds reportedly gathered at the car park on Thursday night.
As of Friday, 165 coronavirus patients had been confirmed to have died while being treated at Russells Hall Hospital.
The Dudley Group NHS Foundation Trust has been contacted for comment.