You see, it isn’t pedantry when still nobody knows what the question is.
I don’t think OP’s question wasn’t about browsing anything, particularly as they won’t clarify.
I think what they actually meant to ask was ‘Can a Watch connect to the internet without my iPhone physically with me?’ to which the answer is yes, as long as you get the cellular version.
I use my iPhone for multiple hours a day for work and play. I don’t have a single issue.