Sure, you have no positive duty to be nice to a child. No-one has an actual job of being polite to random people on the street. Want to make someone kid’s day just that slight bit more unpleasant, to avoid the effort of saying a sentence? Go ahead, there is no law against it.
Take the example of asking a guy visibly wearing a watch “do you have the time?”.
The guy answers “sorry, no”.
It’s not the guy’s job to give you the time, but what would you think of that guy after? Maybe he has a perfectly valid reason for not giving you the time of day. But if he says nothing more, you’d probably think he just doesn’t like your face or something.
For me, it depends on the kid, just as it depends on the other adult who initiates a conversation with me. That’s why I posted my two examples on the previous page. In story one, the mom and kid were polite and respectful and listened to what I told them about interacting with my dog. And everything went fine. In the other story, some random kid comes running at us with a big stick in his hand… that’s not going to end well. A nice polite, “No honey, don’t hit my dog with a stick, you’re scaring her” was obviously not going to slow that kid down one step. And I think he was pre-verbal anyway, but even a kid that young should understand a stern “NO!” in a loud voice. It was obvious a high-pitched “good kid” voice was going to be completely ignored.
So I have no criticism for the OP. We read people as they approach us. There is probably no one single perfect way to handle this situation.
Perhaps I’m assuming too much about people. I would expect an adult to understand that there are many reasons they wouldn’t be allowed to pet a dog.
Also, in the case of the time of day, a reasonable assumption is that the person asking has a need to know the time. Asking to pet a dog is not based on need.
Again, I’ll point out that even rude old me would make more allowances for a child. And I’m generally not that curt anyway about dog petting or the time of day.
And one more little thing to clear up: Did the person (child or adult) making the request use the magic word?
I’d assume that the guy’s watch wasn’t working well, or at all, or he’d forgotten that he was wearing it. Possibly I’d think that he thought I was trying to make small talk.
Sure, no-one needs to pet a dog.
I don’t think whether one gives a curt response to a child really depends on whether that child has asked for something she wants as opposed to something she needs - rather, it should be based on whether that child was acting politely herself when doing the asking.
Nothing in the OP indicated the child had acted rudely when asking to pet the dog.
The expression “he wouldn’t even give you the time of day” doesn’t spring to mind?
As I said, I wouldn’t be curt with a child who was not rude. Children are exceptions.
Thoughts are just that - thoughts. It’s how a person chooses to act regarding those thoughts that determine who they really are.
Hey, I’m one of those pesky people that does occasionally ask to pet other peoples’ dogs. I only do this rarely but I absolutely love pugs, I have one myself, and I love to say hello to other pugs. Usually I only do this if it’s in a setting like a farmer’s market where people are milling around and the dogs are generally well socialized (not flagging someone down on a walk or anything). Or, at the dog park, assuming the dog is leashed and doesn’t just come right up and lean on you for attention. 
I totally understand why there are reasons why you might not be allowed to pet. I get that not all dogs like it. At the same time, if I got a “No” with no explanation, I’d feel, well, a bit deflated. It seems like pleasant social lubrication to give a quick reason, so it’s not a “No, fuck you” or “No, you’re weird” type of rejection – “Sorry, no, he’s a bit afraid of strangers” or “Sorry, no, we’re learning leash manners today” is enough. Maybe it’s my Midwestern sensibilities – I find myself waving at policemen and UPS drivers, smiling and nodding at strangers who catch my eye, and all that. Just seems nice to make it clear it’s not a “No, you weird sweaty dog-manhandling creeper, go away” response.
Now, I wouldn’t go so far to say it’s rude to just say no, it just feels weird to me. It’s like saying “Hi, how are you?” to someone in passing, and them just ignoring you. They’re not obliged to talk to you, it just feels weird, like a brush-off.
The phrase in question in this thread for me is “Sorry, no”. I would have thought that was sufficient. But I would actually say something more in this case, and most others. My rudeness is usually reserved for much ruder questions in real life.
Well, I think that the many people here that disagree with that should be a clue that you and the OP (and few of your fellow cohorts here) two will often be considered wrong and or rude by others that exist in society. Social norms define rudeness, not Vulcan logic.
You guys can bitch and moan about how “nobody deserves an explaination” and so and so on. But guess what? “Sorry, no” is going to be considered curt/rude by many people in this world. You want them to think you are rude? Fine do that. Its your reputation you are putting out there.
But don’t keep acting like society is putting some kind of heavy burden on you by asking you for a one sentence explaination.
Yeah, the mother in the OP reacted rudely. But the OP ain’t winning any awards for positive social behavior either IMO.
This rather surprises me, as I thought “Sorry, no,” was perfectly fine. But I guess it doesn’t hurt any to add a few words of explanation, so this thread has been educational, at least.
My experience has often been the other way around; I’m out walking on a trail, and someone comes up with a dog, and the dog wants to be friends with me! Sniffs my pants, looks up at me with a big happy grin, etc. But even then, I ask the human: “He seems really friendly; can I let him sniff my hand?”
Never assume anything when it comes to other people’s property…especially critters!
I’ll repeat again that I would add some more to a response than “Sorry, no”. And I don’t find it burdensome either. What I’m looking for is some sort of conclusion here, and I don’t see an overwhelming number of people who think “Sorry, no” is rude. I think is still up in the air. Social norms aren’t the same everywhere.
“Sorry, no” can be perfectly nice or curt and somewhat rude depending on the situation.
“Do you know what street the closest bus stop is?” “Sorry, no” is perfectly fine.
“Can I borrow your pen?” “Sorry, no” would be somewhat rude.
I guess it becomes borderline rude when the usual answer is “yes”. Most people who have dogs out and about the general public are happy to have them petted by nice people. It’s ok if you don’t want to have your dog petted, but the expected answer is yes. A word of explanation makes it a courteous response.
Virtually nobody here is considering it a positive though.
Lets see. “Sorry, no”. Some fraction consider it neutral at best, good fraction think “rude/curt/dickish”.
“Sorry, no, explaination X”. Pretty much everbody agrees this is BETTER than just “Sorry, no”.
Choices, choices! How complicated is this?
I’ve always though of myself as pretty socially stunted but dayum sometimes this place makes me feel like I could run a charm school.
I understand that that’s the phrase you’re using, I just have to reiterate that I would find it slightly offputting (though not outright rude). It just seems like the nice thing to do to give a few words why when it’s a pretty innocuous request.
Again, not using, attempting to get a clarification about.
So substitute ‘contemplating, asking about, questioning etc’ into the sentence instead of ‘using’. Answer still works.
Here’s the answer to the question “is it polite to say “sorry, no””?
No.