And a further tweet said: "Esther Curnow needs a punch in the face. That is all."
C4M said one troll urged others to send excrement to the couple, who married in March and live in Newcastle.
They have also been told to "Go and die and rot in hell".
The group said the hate campaign began after Mr and Mrs Curnow took part in the presentation of a C4M petition to Downing Street last week. They posed outside the famous front door in their wedding outfits.
The petition opposing proposals for gay marriage has 575,000 signatories.