According to the Archdiocese of Glasgow, Canon White was spat at, verbally abused, and lunged at.
A 24-year-old man was later arrested and charged in connection with the alleged assault.
The Grand Orange Lodge of Scotland has denied any involvement in the assault on the priest.
Some subsequent Glaswegian Orange walks were cancelled after outcry over the attack on Canon White.
Police Scotland Superindendent Stephen Hazlett told the city council that "this is an area that needs to heal itself. It might take several years to de escalate back to where we were before July 7. We need to give the community time to reconcile themselves and return to normality. The feeling I get is that the time at this present moment is not right. I don't want anyone to get hurt."
A petition at change.org posted after the July 7 attack calling on Glasgow City Council to end the Orange walks has gained more than 83,000 signatures.
James McLean, spokesman for the Rising Star of Bridgeton Royal Black Preceptory said, "We feel we are being marginalised and demonised and that Glasgow City Council are acting clearly at the behest of the Roman Catholic Church."
Scotland has experienced significant sectarian division since the Scottish Reformation of the 16th century, which led to the formation of the Church of Scotland, an ecclesial community in the Calvinist and Presbyterian tradition which is the country's largest religious community.
Sectarianism and crimes motivated by anti-Catholicism have been on the rise in Scotland in recent years.
An April poll of Catholics in Scotland found that 20 percent reported personally experiencing abuse of prejudice toward their faith; and a government report on religiously-motivated crime in 2016 and 2017 found a concentration of incidents in Glasgow.