Mr Wishart, the SNP's longest serving parliamentarian at Westminster, had raised speculation that he might run for the post by calling for a second independence referendum to be put off beyond the next Scottish elections.
Losing a second independence referendum triggered by Brexit would be "unthinkable", Mr Wishart warned last week.
In a blog post on Monday, he said he does not believe he has "sufficient support within the party" and had decided not to stand.
"I said when I was considering standing for the Depute Leadership of the SNP that I would take soundings from colleagues within the party and across the membership before making up my mind to have my name put forward," Mr Wishart wrote on his website.
"After listening very carefully to the response to my agenda, I have decided that I do not believe that I have sufficient support within the party and I will not now be standing."
He added: "I firmly believe that a referendum should take place at the optimum time for success taking into account external features such as the increasing concerns around Brexit, and to proceed only when we have sufficient evidence that it could be won.
"There are certain issues I could have perhaps ducked or de-emphasised in order to better assist me in any depute leader contest, but anyone who knows me knows that this is not something I would be prepared to do. I will always speak out on what I believe is in the best interests of my country."
Mr Wishart concluded: "I hope that others may be able to take up this agenda and perhaps present it more convincingly than I could and I will be asking candidates who do come forward their views on these issues."