Euan Johnston described the Duchess as an "aristocratic minor royal" whose "only qualification is that she sleeps with the queen's son".
His words came after the wife of Prince Charles met with school children from Towerbank Primary in the Capital during her visit to Portobello.
Writing on Facebook, Johnston said: "I am really unhappy about this.
"I think the concept of royalty is insulting to the rest of us and while I accept that the queen holds a constitutional position, this woman's only qualification is that she sleeps with the queen's son."
Johnston claimed the school failed to ask permission if his child could be removed from class to meet the royal.
But teachers have said the children were let out on the basis of start-of-year permissions, which were signed by all parents.
In his Facebook rant, Johnston continued: "I am really not happy that nobody asked me if my child could be taken to 'welcome' some aristocratic minor royal today.
"I also think that in the current political context, after the queen broke her duty of neutrality in the referendum, that attempts at reinforcing Britishness are political."
Johnston's Facebook post prompted dozens of comments from parents and residents.
Janet Jones, who also has a child at the school, said: "I would feel equally appalled and can completely understand that you would be angry and upset by this.
"I had no idea the school had any participation until I saw the post. I'm interested to know did they ask the children if they would like to go?"
Gillian Cummings said: "I think Towerbank is a fantastic school and I've got nothing but praise for the headteacher and the teaching staff, but I was really unhappy with the decision to participate in the royal visit this afternoon.
But a father of another child, who asked not to be named, said that the complainers needed to "lighten up".
He said: "I've got my own reasons, and I expect everyone else has got their own take on it.
"I'm no fan of the royals but the kids enjoyed it and I'm glad the school took the chance to organise something."
Sean Watters, outgoing chair of Towerbank Primary parent council, said many parents would unlikely see it as an issue of concern.
He said: "It's ludicrous to get that level of upset. It's not a big issue.
"As I am concerned, if you're upset about it then you need to find something more important to be worried about."
Education bosses have also said the decision to take the children out of class was in line with current regulation.
A spokesman for the school said: "Pupils involved in the visit were subject to the relevant permissions to leave the school and to appear in photographs, which are signed by parents at the beginning of term."