ETtechByteDance Ltd. has acquired one of China’s largest private hospital chains for about $1.5 billion, deepening a foray into health care via one of the largest domestic tech deals since Beijing’s internet crackdown. TikTok’s Chinese owner paid about 10 billion yuan to take full ownership of Amcare Healthcare, which runs women’s and children’s hospitals in cities from Beijing to Shenzhen, according to a person familiar with the matter, asking to not be identified discussing private information. Two ByteDance subsidiaries now own a combined 100% stake of Amcare, according to corporate registry tracker Qichacha. ByteDance’s health-care app Xiaohe competes with Alibaba…