It comes after former world championship boxer and Alba Party candidate Alex Arthur apologised for mocking the homeless.
When he launched his party on Friday Mr Salmond promised an “entirely positive” campaign with candidates of “strong calibre”.
Dr Walker, who is bidding to become an MSP on the Central Scotland regional list, said he was “pretty mortified” by the now-deleted tweet.
He wrote: “I have removed a tweet which was an inappropriate response to what I thought was mudslinging by the First Minister.
“I freely apologise to her for the comment and also to my fellow Alba candidates.
“We must follow the leadership of Alex Salmond and rise above all negativity as we make the case for our noble cause of independence for our country.”
Dr Walker has previously criticised SNP currency plans and suggested an independent Scotland could use “multiple currencies” in place of sterling.
Writing in Holyrood magazine last December he said: “It is even conceivable – and I for one would highly recommend it – that multiple currencies could be used in an independent Scotland.
“Sterling, the euro, the dollar and the Scots pound (maybe even Bitcoin) could all play a part.”
Mr Arthur, who is standing in the Lothian region, apologised for comparing Romanian beggars to “over fed pigs”.
In January last year he tweeted: “Them Romanian beggars in Edinburgh ain’t hungry!! Just drove past them all ready to switch up begging posts and ALL em fat as big juicy over fed pigs!”
The Commonwealth Games gold medallist also said sorry for a tweet which appeared to undermine the Covid vaccination plan.
Describing himself as a “working-class boxer who shoots from the hip” he also claimed his remarks had been “misinterpreted” by some.
George Kerevan, the former SNP MP for West Lothian, also announced he would be resigning from Ms Sturgeon’s party and joining the Alba.
He made the announcement alongside the founder of the Common Weal Group (CWG) in the SNP, Craig Berry, and the SNP’s central Scotland coordinator, Lynne Anderson.
In a joint statement they described Mr Salmond’s outfit as a “cause for hope”.
Brexit-backing SNP Aberdeenshire councillor Brian Topping also revealed he was switching parties and would stand for Alba on the North East list.