Agutyar, obviously it depends on the society. If calling a person by his/her first name implies closeness it is a problem. However, today in most situations (an exception is if one is the boss and there it also applies between people of the same gender) it does not.
Regarding Rav Moshe’s pesak, Rav Eli Mansour says that “the halachic authorities” have ruled that there is no problem and then cites Rav Moshe’s opinion. You can ask him on-line what he means. In any case, even Rav Moshe is only machmir lechatchila. Certainly if it would cause ill feelings, she would be stranded, etc. he would be meikal.