The man attempted to obtain a court order which would have barred his wife from sending their "very bright" children to more mainstream Jewish schools where he feared they would lose contact with the traditions and community in which they had been brought up.
He objected to the prospect of his daughters being sent to a mixed-gender school and to his children having unrestricted access to television, cinema, "certain newspapers", the internet and social networking sites, the court heard.
During their 10-year marriage, which ended two years ago, the couple observed the traditions the conservative Chareidi Jewish community in London.
Their observances precluded them switching on a light, catching a bus or making a phone call on the Sabbath.
But after studying for an Open University masters degree, the mother has forged her own career and wants her children to have the schooling she says she never had.
Eleanor Platt QC, representing the father, told the court that he was deeply concerned that his former wife would lead his children away from their community and denigrate his lifestyle in their eyes.
She said that the children are thriving at their current schools and moving them away from the lifestyle would cause emotional harm by separating them from friends they had known all their lives, who might in turn be forbidden from associating with them.
Miss Platt added that the mother now lives a long walk away from the nearest synagogue, a serious problem in traditons in which even use of a pushchair is forbidden on the Sabbath.
But the mother told the court that educating the children at an ordinary Jewish school would give them "infinitely superior opportunities".
The Appeal Court judges rejected the father's application for a joint residency order and for the mother to be refused permission to move the children to new schools.
Lord Justice Maurice Kay, Lord Justice Munby and Sir Stephen Sedley will give detailed reasons for their decision at a later date.