No. If the baby is born in America, China government will not care about that.
她可以去美國的醫院做各種產前檢查嗎?
If she has enough money to do that, she can. However, her new baby will not be the excuse to let her to stay in America or change status until her baby to become a adult to help her to enter America.
不會因為她拿的旅遊簽證在美國生小孩,被移民局發現把她驅除出境?
Could be.