Trevor Mallard - the MP whose tweets against the Speaker sparked a parliamentary inquiry into the use of social media by MPs - says there should be no special rules for social media.
"In my opinion there is no difference between making a criticism on new media on Twitter, to making it to a TV camera, to making it in the House," Mr Mallard said.
"While I understand there is a review going on, my view has always been that you are subject to the privilege rules whatever media you use."
Submissions for the inquiry by the privileges committee close today.
After disagreeing with a ruling Speaker David Carter had made during question time last May, Mr Mallard tweeted:
Education spokesman Chris Hipkins accused Mr Carter in a tweet of applying a "double standard".
MPs regularly criticise each other on Twitter, and have taken pictures of other MPs reading or slouching in the chamber and of the public galleries, which are banned activities for news media.
The tweets by Mr Mallard and Mr Hipkins are prima facie privilege breaches, questioning Mr Carter's impartiality. Breaches of privilege are not confined to what is said or done in the House but apply equally in any medium.
It's acceptable to criticise the Speaker in his role as Minister of the Parliamentary Service but not to criticise his impartiality or competence.
National complained after Mr Mallard's tweet and Mr Carter referred the matter of social media generally to the privileges committee.
Mr Mallard is now an assistant Speaker and says it would be "exceptionally unwise" to be tweeting similarly now.
As for using social media to criticise other MPs, he said: "I see it no different to a similar discussion ... on talkback or in the comments section of the Herald."
National MP and tweeter Judith Collins said rules against criticising the Speaker should apply to social media as much as other media.
Social media review: Terms of reference
• Should any restrictions or guidelines apply to members' use of hand-held electronic devices in the Chamber and committees, including accessing social media to comment on the proceedings?
• Do Parliament's procedures and rules need modernising to reflect the opportunities and challenges provided by social media, including where the House and committees use social media to disseminate information and facilitate public participation?
• Are the current rules that reflections on members (including the Speaker or other presiding officers) could amount to a contempt; or accusations against the Speaker that reflect on him or her in that capacity could amount to a contempt appropriate?
• If so, should these rules also apply to reflections or accusations made outside the House or on social media?