Midwife Caroline Waller visited their home on September 15, three days after the birth of their twins Oliver and Harry, for a routine check-up.
The midwife says she tripped over a buggy which was on the stairs, suffering a back injury, as she was leaving their home.
However, the dismayed couple said she merely stumbled and that the single buggy was folded away and at the bottom of the stairs.
The Hudsons, from Hull, East Yorks, said the arrival of their babies had been 'tainted' by the incident.
Oliver and Harry were born two weeks ago and - at 9lb 6oz and 8lb 5oz respectively - are believed to be the fourth largest twins to be born in England.
Mrs Hudson, 34, said: 'It's stressful enough having a baby, let alone twins, but this is even more stressful. There is a real blame culture.'
IT director Mr Hudson added: 'Most people just claim sick, don't they. That's fair enough if they have hurt themselves.
'But this? If our one-year-old leaves something on the floor and a friend trips, will they sue us too? I think it's ridiculous.'
On September 22, they received a letter from Thompsons Solicitors, who act on behalf of the union and professional body The Royal College of Midwives. The letter threatened legal action under 'The Occupiers Act 1957.'
The couple believe they will be covered under home insurance but fear their premium will go up - and do not want to pay out on principle.
Thompsons Solicitors declined to comment on behalf of the firm and their client.
The Royal College of Midwives said it could not discuss individual cases.
Although the midwife is employed by Hull and East Yorkshire Hospitals, the NHS trust is not involved in the case.
Ms Waller could not be reached for comment, Daily Mail said.