I realise Torchwood was formed partly because the Doctor can't always be there, but sometimes events are so big it is strange that he doesn't turn up (eg Miracle Day). Can anyone suggest why he might not?
Torchwood was formed to defend against the Doctor (10th Doctor episode Tooth and Claw), not to work with him. It's only in the final days of the Torchwood Institute that they aren't predominately working counter to the Doctor, so there's not a lot of incentive for him to be stopping by to help them out.
In the later days, of course, there was more attachment, with various companions working for/with Torchwood. However, we've seen that once a companion leaves the Doctor, he moves on, and this is especially true once he regenerates. We've seen mixed feelings towards Jack in particular - he often doesn't agree with the way Jack does things, and he's not totally comfortable with the fact that he ended up immortal through their time together (with Rose).
It seems reasonable to assume that the 11th Doctor doesn't feel any special need to assist Torchwoord, so it would only be if he wanted/needed to help Earth itself. However, the Doctor doesn't solve every problem in the universe (he's busy enough solving the ones he caused!); let alone all the ones that exist on Earth.
Also, from all appearances, most of the time when the Doctor shows up and solves a crisis, it came about by accident - he happened to turn up at a fortuitous time. We see a few deliberate journeys, but they are in the minority; it's possible that these aren't as accidental as they appear, of course, but the evidence is that he fixes things as he finds them - he's not Sam Beckett seeking to right what once went wrong.
So he's not really keeping an eye out to help Torchwood, and he didn't happen to blunder into Miracle Day, so the question is whether he would have been called to help somehow. Even if he was, since the 9th Doctor, we've had many mentions of "fixed points in time":
Fixed points were events and/or individuals who had such long-standing impacts on the timeline that no one, not even Time Lords, dared interfere with their natural progression. The Doctor, free to interfere in alien invasions and save planets in most cases, could neither interfere nor interact with these fixed points. Were a fixed point to be interfered with, the change would be circumvented, making the timeline continue despite changes.
It's possible that Miracle Day (and other things Torchwood has dealt with) was a "fixed point". (However, Jack does say that the future is "still being written", and if it was a fixed point, there's a chance that he would know about it given his history). If it was, then there would be nothing the Doctor could do anyway.
Finally, it's possible that the Doctor was involved, and we just never saw it. He's had around 1,000 years of life (and presumably more that we haven't seen anything of yet) and we only get glimpses of small parts of this. Perhaps the Doctor (any incarnation, really) nudged a few vital pieces into place, without Torchwood (or us) knowing anything about it, while he was having some other adventure on Earth in mid 2001.
If you want an in-universe answer, there's a monologue from Gwen in Children of Earth where she says, "Sometimes the Doctor must look at this planet and turn away in shame."
In real life, Russell T. Davies stated he wanted the two to diverge more and more and, after the last crossover with Jack appearing in Doctor Who and following that with Martha Jones appearing on Torchwood, said more crossovers were unlikely since the two shows were going in different directions. Davies said, at one point, that one thing he liked was being free to kill off regular characters, and he felt he could not do that on Doctor Who, since the Doctor essentially had to be able to rescue everyone at the end.
Torchwood has an entirely different tone than Doctor Who. Problems on Doctor Who are solvable, and can often be solved without ugly choices, whereas problems in Torchwood are darker, deal with a much darker aspect of the human psyche, and the solutions are usually quite messy. Note, for examples, the sacrifices Jack made when dealing with the child chosen by what we think of as faeries (Small Worlds) and in Children of Earth. The intent is that the problems in Torchwood can't be solved easily by someone like the Doctor so everyone is okay at the end of the story.
In many cases, like Children of Earth and Miracle Day, if the Doctor appeared, he'd be providing a quick solution and characters would not be making dark and serious decisions and facing the consequences of those decisions. That's the darker tone Davies intended Torchwood to have.
So if you're looking for a simple answer: While nothing is impossible, appearances by the Doctor on Torchwood are extremely unlikely since the two shows are so different in tone and growing apart more and more.
There is an argument as to why he does not solve all of the earths problems in one go. The fact is that he does some stuff, and he saves the earth sometimes from some major threats, but not always, and not from everything. Torchwood also saves the world from some things, and not others.
One could ask, why doesn't he solve world hunger, peace and global warming. He is not earths deus ex machina to solve everything. He is a timelord with a particualr interest in earth. Sometimes.
I agree and disagree with some of the above comments. One thing I disagree with is labeling Doctor Who a children’s show. I’m not sure I’d let little kids watch Doctor Who as it more mature than people give it credit for and here in the US it airs as an adult show though the sexual references alone make Torchwood obviously for mature audiences. I believe Gwen’s comment regarding how the Doctor must sometimes view humans rings true and I think that he trusts Jack. Torchwood was originally formed after a…royal encounter with the Doctor, but its mandate is to defend against extraterrestrial threats in general. When the Doctor first encountered Torchwood during the whole cybermen incident they did indeed capture him, but they did so to acquire his technology and himself as sort of a living weapon/tool against other extraterrestrials. As for my reference to trusting Captain Jack what I mean is while his being a fixed point in time makes the Doctor uncomfortable due to his ability to see potential futures, travel through time, etc… he has shown that he considers Jack a friend and the only other person who is likely to live as long as him due to his immortality. I know the Doctor prefers less violence but he knows it is sometimes necessary so in the end leaves a very capable Torchwood to the dirty business as I’m not sure some of their situations could be solved by diplomacy. I’m sure the Doctor would work to destroy Torchwood if he didn’t think it was needed though I think the fact it is run by Captain Jack has something to do with that decision. In the end if Torchwood truly faced a world ending situation they couldn’t solve I’m sure the Doctor would help. The main reason he doesn’t show up reality based wise is obviously because it is supposed to be a separate show and they don’t want conflicts between the two in regards to viewers/ratings not to mention it would take time away from filming Doctor Who to have him constantly guest star on Torchwood, though unless something changes that’s a moot point at the moment with no Torchwood.
Just a random thought haven't really thought it through completely. But what if because Jack is a fixed point in time then anything he is involved in is considered fixed? Therefore the Doctor would be unable to help in things like Children of Earth and Miracle Day because Jack is a part of it which causes them to be fixed.