Is the fast of one who vomited unintentionally on Friday—while observing the six days of Shawwal fast—valid, given that he completed his fast and fasted the following day?

Your fast is valid, and vomiting does not harm it if it was unintentional or involuntary, because whoever vomits unintentionally, their fast is valid. As for whoever vomits intentionally, they have broken their fast. This is based on what Al-Tirmidhi narrated from Abu Hurairah (may Allah be pleased with him) that the Prophet (peace and blessings of Allah be upon him) said: "Whoever is overcome by vomit, there is no make-up on him. But whoever vomits intentionally, let him make up for it." This is what the general body of scholars has adopted, and whoever is overcome by vomit, it does not harm them, nor does it break their fast.