A Liberal Democrat motion urged the council to launch a working group to the terms of reference and objectives for the enquiry and which other organisations should take part.
It was supported by the Labour group leader, Dr Julia Buckley, at Thursday's full council. But councillors voted instead for looking into the feasibility of such an enquiry.
Putting forward the motion, Councillor Ruth Houghton said everyone knew of the daily ambulance delays within Shropshire.
"I have seen ambulances stacked more than 10 at a time outside the Royal Shrewsbury Hospital, doors open and people being cared for inside. Shropshire residents are worried and frightened.
"If ambulance attendance times are being missed in our market towns, it is simply impossible in our rural areas," she said.
Councillor Roy Aldroft said the problem was not with the ambulance service but rather with the inability of hospitals to admit patients because of a shortage of staff and bed blocking.
"The problem lies with Shrewsbury and Telford Hospitals Trust and on the social services side when it comes to discharging patients," he said.
The amendment to consider the feasibility of an enquiry came from cabinet member, Councillor Dean Carroll who asked: "Should it be the council leading this enquiry, is this the appropriate avenue. We have already had meetings with our counterparts in Powys and have had long standing conversations with the health service."
Supporting the motion, Councillor Buckley said there had to be cross-party concern and action.
"There is no place for party politics when it comes to saving lives, when handover delays mean queues of ambulances behaving like wards.
"Everyone suffers with there are handover delays but the geographic differences mean that the response times go up for our market towns and rural areas.
"The rural areas are collateral in this crisis."
Figures read out by Councillor Bernie Bentick showed that in April 72 per cent of ambulances in Shropshire did not meet their response targets.
"Many people in hospital are due to be discharged but have neither a bed in the community to go to our a care package in place for them," he said.
Speaking on the amendment Councillor Roger Evans said: "We need to get this sorted now, not look at the feasibility of it. My 96-year-old neighbour had a fall and broke his hip. It was 14 hours later that the ambulance arrived and 14 days later I went to his funeral. Our residents are suffering."
Council leader Lezley Picton said: "What is an enquiry going to achieve and what is it going to cost. It will tell us the health system isn't working - well we know that."