Q. A Ger who converted many years ago was told not to maintain constant contact with his parents as before his conversion. Now that his mother is aged, ill and alone and requires continual care, and no one else is left to care for her, can he maintain constant contact?
Is it advisable to bring his children to visit their grandmother, if she is pained by not seeing them?

A. The Talmud (Yevamos 22a) teaches that although a ger who converts is considered as a new born person and looses his familial connections to his biological family, rabbinically he cannot marry his sister or other close relatives. The Talmud explains that the reason is, people will say the ger left a stronger sanctity and converted to a lesser and more lenient religion.
Rambam (H. Ma’amrim 5: 11) rules that not only should a ger observe the negative commandments in regards to dishonouring or shaming his parents, but he should also show some honor and respect to them. (See also Shulchan Aruch Y.D. 241: 9) Yad Avrohom explains that honoring parents is an obligation for Bnei Noach, as we can infer from the narrative of Chom Ben Noach, yet they are not punished for non-compliance as they are in the case of the other seven Noahide mitzvos. Rabi Akiva Eiger in his notes (ibid.) maintains that it is referring to the way Gentiles in their own beliefs and laws, greatly honor their progenitors.
Igrois Moshe (Y.D. 2: 130) in a similar shaile rules that even when the parent is not ill, the ger is permitted an occasional visit and is also allowed to bring along his children. However, he should do so infrequently so that the children will not adhere to the former religion of their father, join his grandparents in eating non-kosher and do other common prohibitions.
Otherwise, in case of need or illness, just as one has to take care and visit the afflicted gentiles and tend to their needs, certainly and more so, the ger has to take care of his own parents. (See also Nishmas Avrohom 2: p. 276)
Horav Shlomo Miller’s Shlit”a opinion is similar.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a