The SNP deputy Westminster leader labelled the report a “damning indictment” of the Prime Minister after the document criticised some of the language he used.
Mr Johnson was cleared by a majority on an independent panel over a complaint he broke the party's code of conduct following a Daily Telegraph column in 2018 that described Muslim women who wear the burqa as looking like "letterboxes" and "bank robbers".
However, the Singh Investigation report said several interviewees who spoke to the inquiry considered Mr Johnson's language "discriminatory and unacceptable".
In a pointed message to Mr Johnson, the review led by Professor Swaran Singh said the leadership of the Conservative Party "ought to set a good example for appropriate behaviours and language".
Responding to the report, Ms Oswald said: "This report is a damning indictment of Boris Johnson’s conduct and the behaviour of the Conservative party more generally.
“The fact that the Prime Minister’s remarks have been found to have given an impression that the Tories were seen as ‘insensitive towards Muslims’ should come as no surprise at all to anyone, in the party or outside it.
“Boris Johnson has made a litany of overtly racist and discriminatory comments about various groups of people over the years, which have not only caused major offence, but have damaged the Conservative Party and the UK's reputation.
“This is not leadership – and any Prime Minister who does not hold, promote and follow a zero tolerance towards racism and Islamophobia is not fit for office.
“This latest finding serves as another reminder of the rot at the core of this Tory government and its extreme, right-wing views."
The inquiry found anti-Muslim sentiment was seen at local association and individual levels, but claims of "institutional racism" were not borne out by evidence of the way complaints were handled.
Mr Johnson has said he was "sorry for any offence taken" over the language used.
He said: "Would I use some of the offending language from my past writings today? Now that I am Prime Minister, I would not."
In response to Mr Johnson's assertion he would not make such remarks now, the report said: "While this could be considered leading by example, the investigation would like to emphasise that using measured and appropriate language should not be a requirement solely for senior people, but ought to be expected throughout the Conservative Party."
The investigation also examined the controversial and unsuccessful mayoral campaign of Zac Goldsmith.
The report said Lord Goldsmith "accepts poor judgement in the way his campaign was conducted, but forcefully denies harbouring anti-Muslim sentiments or using such sentiments for political advantage".
The report said high-profile cases like that of Mr Johnson and Lord Goldsmith "give the impression to many that the party and its leadership are insensitive to Muslim communities".
Prof Singh said: "I'm not saying that the party leadership is insensitive to Muslim communities. I'm saying that the perception is very strong."
The inquiry carried out by Prof Singh, a former commissioner at the Equality and Human Rights Commission, was established following a series of allegations about Islamophobic behaviour and was widened to consider all forms of discrimination.