This comes after Smith branded the Brexit party called it a “shell company that’s a money-laundering front.”
Smith, an ardent Remainer in the Brexit debate who was re-elected in last week’s European elections as the SNP's lead candidate for the Scotland constituency, is accused of defaming the Brexit Party and its chairman, Richard Tice.
He was given until noon on Thursday to respond to the allegation. But he has refused to apologise and called for an independent inquiry into how Farage and the party are funded.
On Thursday, ahead of the deadline, Smith told this website that said he had referred the firm to the case of Goldsmith and Another v Bhoyrul and Others - a 1997 case which established that political parties do not have the power to sue in defamation proceedings.
He said, "Like many others, I would like to see a full, open, and transparent independent inquiry into the funding of Nigel Farage and the Brexit Party."
The deputy made the remarks in a television interview in the early hours of Monday as the European Parliament votes were being counted.
The Brexit Party finished second to the SNP in Scotland, with 14.8 percent of the vote, compared to the SNP’s 37.8 percent.
"Like many others, I would like to see a full, open, and transparent independent inquiry into the funding of Nigel Farage and the Brexit Party" Alyn Smith MEP
During the Sky TV interview, Smith described the Brexit Party as a "money-laundering front".
Brexit Party Chairman and newly-elected MEP Richard Tice instructed lawyers to contact the politician demanding he retract the "false and highly damaging" claim.
Tice instructed legal firm Wedlake Bell to contact the MEP demanding an apology.
A letter, seen by this website and sent to Smith from the firm, said that "any viewer who was aware that Mr Tice was the chairman of the Brexit Party would conclude that you were alleging that Mr Tice is himself running a money-laundering operation".
They added: "Your allegation is categorically denied. The Brexit Party has not acted as a money-laundering front and Mr Tice has no involvement in money laundering or any other law breaking.”
"Mr Tice is deeply affronted by the accusations you have made against him and the Brexit Party. Clearly your political viewpoint is different from his, but that can be no excuse for making such a very serious and totally unfounded charge of criminality against our client."
The firm said that unless Smith "set the record straight" before noon on Thursday, "Mr Tice will have no option but to instruct us to bring High Court proceedings against you.”
Wedlake Bell said the interview has been reproduced in numerous tweets, including one which has been viewed more than 27,000 times.
Smith has been particularly vocal of the Leave camp in the Brexit debate in the UK.
The Brexit party won 29 of the UK’s 73 seats in last Thursday’s elections in the UK. It was formed by Farage in November 2018 with the aim of ensuring the UK’s departure from the EU.