Drinking laws are provincial matters. Say you're 19 and you're from the USA so you're not able to drink there legally but you go to Canada, would you legally be able to be drinking there? Surprise, surprise, little-travelled American young person. Yes, at age 19 you can legally drink alcohol in any province in Canada.
If you're talking about US style grocery store alcohol purchases, where you can buy just about anything, no you can't do that anywhere in Canada other than Quebec. Ontario and New Brunswick both allow an extremely limited selection of beer and/or wine to be purchased at grocery stores.
In short, yes, if you are intoxicated at a level where you may pose a danger to yourself or others around you. The determination on whether you are so intoxicated or drunk in public that you pose a danger to yourself or others is left initially to the arresting officer.