I guess the most-likely place for the answer would be in early source materials (the widow theory probably came from somewhere). There’s at least two other scripts out there, so the answer might be there.
If it were to ever be addressed I think it’d most likely be in association with the first script which was written before 1995. This is just me, but I feel like it’s unlikely that it was ever canon-ly covered. His mother was probably always referred to simply as ‘a single mother’ if that was even mentionable at the time because it’s possible they choose to keep discrete. Most of the movies that start and end with an unmentioned but missing parental figure stays unmentioned. Aladdin’s mom, Remy’s mom, Jane’s mom, Belle’s mom, Jasmine’s mom, Snow White’s mom, Dumbo’s father…
I have no doubt writers have very very personal and private back stories for these missing adults and there are plenty of inside joke theories too, but nothing that was used in the creative process of the movies as far as reasons why they were cut or not written in at all.
No, I realize this happens a lot in the world, fictional or otherwise, I just seriously doubt that a subject of this nature would be even considered in Pixar, let alone Disney.
But I just gave you an example that it did 
I unfortunately looked over that little bit. I apologize.
Let me rephrase - I don’t think that Pixar would cover something like that often. We all know they can handle mature matters like that, but I don’t think they’d wanna open a can of worms like that often.
I personally don’t think Pixar has any qualms with worm cans. x] In my opinion they don’t mind at all as long as there is a reason and it furthers the plot. In Ratatouille it was involved in a big turn in the plot. For Toy Story a back story wouldn’t add to the progression of the film. Thus why the subject isn’t covered at all.
Either way Pixar doesn’t seem to cover said issue at all! Because it’s not something to be covered, it’s just life.
From the beginning, Woody was supposed to be a handmedown from his dad, so they were probably close at one point. It’s almost apparent that he died, because like I said earlier, he missed a lot of events in his kids’ lives.
I love the plot that Woody belonged to Andy’s father. But this information was not released until long after the first movie came out, and I think it was an after thought. Just my theory And even if it was not, I think it still wasn’t discussed what happened to his father.
I didn’t think Molly and Andy being born out of wedlock was bad, just an idea. She just seems like the type to me…
It really isn’t a rule that you have to be married to have a child. Me and my brother were born while my parents weren’t married.
No one said it had to be.
Do you mean to say you think this is also a good theory that you stand behind as far as where Andy’s dad is? Or what is your theory?
…what are you talking about? I was replying the girl above me.
EDIT: I’ve already posted what I thought.
I’m talking about that she didn’t say that xD
Sorry if it bothered you that I responded, I wanted to make sure you weren’t offended or anything :]
Where did I indicate that she did?
It was just a statement.
I wasn’t.
GOOD GOOD! I’m so very glad. That would have been terrible.
Okay…? 
Personally, if Andy’s dad is alive, I think my idea is right. I’m not against other ideas, though.
I’m sticking with the idea that his father passed on.
Yeah, me too. 
?
If he were alive, I think maybe I would think this too. Though I think I lean more towards the possibility of a divorce.
My personal theory is that he passed.