The man”s estranged wife told Judge Judith Rowe of her concerns about what the children were allowed to do when they visited their father.
She said she was afraid of them being “exposed to an alien way of life” and of religious rules being broken.
The mother said the children had “made many allegations”.
One child had been allowed to press a traffic light button on the Sabbath.
The other had been “shown an electronic device”. They had also had been allowed to eat non-kosher food, she said.
The woman also accused her estranged husband of taking off his kippah on Sundays.
Detail of the case has emerged in a ruling by Judge Rowe following family court hearings in London. No-one involved has been identified.
Judge Rowe said family court litigation had started more than two years ago.
She said she had made decisions relating to when the children, who lived with their mother, should spend time with their father.
“The whole family lived a Satmar ultra-orthodox Jewish life until early 2013 when the father decided to leave the community,” said Judge Rowe in her ruling.
“This was a seismic event for the family. The father continued thereafter to follow his Jewish faith but at least for a time he did not practise within a settled Jewish religious community.
She added: “The court had to contend with the mother”s distress and fear of the children being exposed to an alien way of life on the one hand, and the father”s impatience at having to continue to follow the Satmar way of life during his time with the children on the other hand.”
Judge Rowe said she did not think that the man had deliberately flouted “expectations on him”.
She said “two loving parents” were struggling with arrangements for the children”s care.
She said the youngsters should spend equal time with each parent.
And she said it was important that the children saw that their father was “still Jewish”.
Judge Rowe added: “There have probably been occasional deviations from the Satmar rules for reasons including mistakes and occasional carelessness of the father or others, however I do not find that the father has deliberately or intentionally flouted the expectations on him or, in any event, that these have been either frequent or generalised.
“The father knows what is at stake here. I find that he would not now deliberately cause problems for the children.
“They must spend important religious days with him just as with the mother. Both parents must be part of the fabric of the children”s lives.”
Judge Rowe said the children had to be given the “clear message” that their lives with each parent were “equally important”.