Is there a regulation in the Army about wearing utilities in public in a non-duty related activity?

Define "non-duty related activities". It is permissible to run quick errands while in uniform, particularly on the way home from work. It's also permissible to eat meals in uniform provided the establishment is not primarily involved in selling alcohol. This is still METT-T and post policy dependent (as well as common sense), but generally, since the start of GWOT, the Army has taken the attitude that we are at war, and our citizens should see our soldiers, and have relaxed a lot of the old regs regarding off-post wear of the uniform.

Now if you are talking about wearing ACUs on the weekend while not on duty, or at an off-duty public gathering, then that is a no go.

Actually, it depends on the non-duty related activity for ther Army. Army Regulation 670-1 and AR 600-20 as well as local post regulations all govern when the wear of the uniform is authorized and prohibited. For example, AR 600-20 deals mostly with times when wearing of the uniform is prohibited in order to prevent the appearance of the endorsement of political or ideological groups that are not officially endorsed by the US Army.

There's no rules against it, however you shouldn't go to happy hour at Applebee's for instance and drink alcohol