If I filed the petition for Alien Spouse, I-130 form, while he is visiting in the US, does he go home to Italy or does he have to stay in the US?

Significantly more information is needed in order to provide a legal analysis of ways your husband may be able to attain immigration benefits through a marriage-based application. This includes, for example, information about your own immigration status (Are you a U.S. citizen? Are you a Permanent Resident? Do you have some other status?); information about your husband (How and when did he enter the U.S.? What is his current immigration status? What immigration-related applications every before have been filed by or for him?); Etc. There really is no substitute for you and your husband to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.