Councillor Malcolm Pate says that the authority's own surveys have shown little public appetite for a rate hike, and that holding a referendum would be "throwing good money after bad".
The Conservative council leader says that a county-wide poll on the issue could cost as much as £700,000, with no guarantee of a win.
It comes as reports have emerged that Surrey County Council is considering holding a referendum on increasing council tax by as much as 16 per cent in a bid to tackle ballooning costs of providing adult social care.
Councillor Pate, who has previously rejected calls for a referendum, said: "I think it is highly unlikely. We did the Big Conversation and that did not show a majority of people wanted council tax increases.
"And it is quite costly. It could be £600,000 to £700,000 to hold it and if we are not confident we are going to win it then that is just throwing good money after bad.
"What we will be throwing our efforts into is making government understand the costs of delivering elderly care in a rural county. I think the government will have to acknowledge that the formula is not sufficient.
"We have made it quite plain to central government that the formula used for elderly care is ridiculous and based on business rates and council tax."
Last month Councillor Pate wrote to local government minister Sajid Javid to call for greater funding for the county to provide adult social care, warning that the current system was leaving the authority "rapidly reaching a point where we will be unable to afford adult social care".
Government proposals, unveiled in December, would allow councils to add a three per cent council tax rise solely for use on adult social care in each of the next two years. Currently councils are allowed to increase council tax by two per cent a year solely for adult social care.
Under the new proposals the increase will only be permitted if councils collect no further rise in the third year.
Councillor Alan Mosley, who leads Shropshire Council's Labour group, says the onus should fall on government to provide sufficient funding.
He said: "I think the government policy is appalling in that they need to recognise a national issu. This is a government problem and they should be handling it through proper grants."