Can he contact his (female) cousin to greet her by phone?

Praise be to Allah.

There is nothing wrong with your husband speaking to his (female) cousin and asking how she is, so long as he pays attention to the shar’i guidelines concerning that, i.e., there should be a need for him to speak to her; there should be no softness in speech on the woman’s part or enjoyment thereof on the man’s part; the talk should be permissible and serious, with no joking or laughing. His cousin is a stranger (non-mahram) like anyone else, so she should not be soft in speech when talking to her (male) cousin, or talk too much to him for no need, or mix serious talk with jokes and laughter and the like, which is something that people who do not know the shar’i limits concerning that usually take lightly when talking to relatives. If the man wants to check on his female relatives and see how they are, there is nothing wrong with him greeting them, asking how they and their parents and children are, then he should end his conversation. This may be done at various intervals, whenever there is a need for that.