The Glasgow Orange Defenders face potential action from Northern Ireland's Parade Commission amid allegations they had been provocative by continuing to play as they passed St Patrick's RC Church on Donegall Street in Belfast.
It's claimed the incident happened during twelfth of July celebrations.
The Commission has a rule against flute bands playing music outside churches during parades.
Alban Maginness, member of the SDLP party, wants sanctions against the band.
He told the Glasgow Herald: "It would, in my view, be appropriate to ban them from any further participation in parades in the area or indeed in Northern Ireland at large.
"There are huge historic sÂensitivities around St Patrick's and this band is in a situation where they can carry on and no-one takes much notice of them back in Scotland.
"They've a freedom they wouldn't have in Scotland and they've exploited that freedom to make a statement. But it would be a wrong assumption to make that if you're back in Scotland you're virtually untouchable."
The band sergeant posted a statement on Facebook: "If playing a hymn passed [sic] a Christian place of worship is wrong then we must be guilty. If asked if I would change or play anything different? Never. We are what we are."
Marchers say they performed Land of Hope and Glory and Abide with Me during the parade.