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

I guess this is one case where the Air Force has something in a regulation and the Army doesn't.

AFI1-1, Air Force Standards
3.4.2. Prohibitions on Wear of Uniform. Air Force members will not wear any uniform combination or any uniform items in the following situations:
3.4.2.1. When attending a meeting of, or sponsored by, an organization, association, movement, or group that: the Attorney General of the United States has named as totalitarian, fascist, communist, or subversive; advocates or approves acts of force or violence to deny others their rights under the United States Constitution; or seeks to change the United States government by unconstitutional means.
3.4.2.2. When participating in or attending public political speeches, interviews, picket lines, marches, or rallies, or in any public demonstration when participation might imply Air Force sanction of the cause or if the purpose may be to advocate, express, or approve opposition to the Armed Forces.
3.4.2.3. When it would discredit the Armed Forces.
3.4.2.4. When furthering political activities, private employment, or commercial interests.
3.4.2.5. When engaged in off-duty, civilian employment.
3.4.2.6. When participating as a defendant in civilian court proceedings if a conviction would bring discredit to the Air Force.
3.4.2.7. Air Force members may not wear distinctive uniform items with civilian clothes. Distinctive uniform items are those items that are unique to the uniform, such as grade insignia, ribbons, cap devices, badges, uniform jackets (not to include the PT jacket), and other United States or Air Force insignia.
3.4.2.8. When eating at off-base restaurants where most diners wear business attire, or at establishments that operate primarily to serve alcohol, Air Force members will not wear the Airman battle uniform (ABU) or flight duty uniform.
3.4.2.9. When using frequent flyer miles to upgrade to business or first class, Air Force members may not wear military uniforms. Even when an upgrade is legitimate, wearing of the uniform may create the public perception of the misuse of government travel resources; therefore, wear of the uniform should be avoided under these circumstances.

I went to the shopette, commissary, gas staion, Class 6 etc in my uniform, usually because I was on the way back to the hooch afterwards.

When I was ADSW at YTC, I frequently went shopping, picked up my kids, etc. out in the civy world while I still had my BDUs on.

I never put them on simply to go out and about, I more often just didn't bother to change into mufti on while on my way to home or wherever AFTER my duty day had ended.

I have found nothing that bars the wearing the duty uniform off duty while shopping, etc. I do stop at certain quik marts for gas or a pack of smokes and continue on my way. I do not go down town, grocery shop, or do any lengthy shopping in my duty, or for that fact, dress uniform. It attracts attention that I do not want either favorable or unfavorable. My local command highly suggests and encourages not being in uniform for shopping or dining while off duty simply due to the possible effects of attracting negative or targeted attention. In today's world, given our current adversaries, I am of the mind of safety and attracting attention to ones self while in uniform out of the realm of relative safety of post or garrison is a risk I try to lower. I am VERY proud of my uniform and what I have done to earn the right to wear it over the last 17 years of service but sometimes you just gotta play it a bit smarter.

I don't understand why the other services have not put much effort in to uniform regulations. I found a 65 page thing for the Army; but it's not as detailed as it should be. That better than nothing, though. If you just Google uniform regulations, the Navy is the first that comes up with a myriad of information that will also allow you to easily navigate your way to the uniform information you need. I've seen Army uniforms that I believe to be incorrect; but I have nothing to stand on without a easily accessible regulation. Maybe if there is someone on here with any sort of influence, maybe the idea can be brought up the chain. It would benefit every service, even if they're not in the same service.

Short answer, No.

I seen this back when I was at Langley AFB and had seen guys in town(Army) wearing
Fatigues In the AF when I was in, a person was permittted to make a quick stop get what’s needed, pay and get going. No hanging around... hit Burger King.. do drive through.. what I would have done, don’t like it but it’s quick!
The regulations for you guys are there, just read and find out! Just my nickle’s Worth !

I would refer you to AR 670-1 Wear and appearance of the Army Uniform. I dspentn24 years as a soldier and so much so, I was not interested in wearing my uniform or any of my gear off duty!