Alliance Party councillor Claire Kemp called on Catney to explain his “rationale”.
She said many residents had expressed “deep unease at anything honouring Andrew Mountbatten”.
“Lisburn and Castlereagh has a long and proud history with the Royal family but it was entirely correct for the council to remove the plaque mentioning Andrew Mountbatten,” she added.
Catney and the SDLP were approached for comment.
An SDLP spokesman said: “Councillor Catney has already clarified his position to council and has stated that he believes the plaque should be permanently removed.
“The SDLP has been clear from the outset of the Epstein scandal that tributes to Andrew Mountbatten-Windsor are inappropriate, and we were among the first to call for their removal.
“Our thoughts are with all the victims of Epstein and those associated with him.”
A spokesperson from Lisburn and Castlereagh City Council said: “The mount has been removed as part of a site tidy up and there are no intentions to replace at this time.”