TL;DR: That's not how fact checks work.
Politifact has a Trump page which as of right now (October 4th, 2016) says
Mostly True (11%)(31)
Half True (14%)(39)
Mostly False (18%)(48)
Pants on Fire (18%)(48)
If you do the math, this gives 71% Mostly False or worse (False and Pants on Fire) versus 29% Half True or better (Mostly True or True).
So on the face of it, the 22% statistic is currently wrong, although it may have been facially correct at the time it was originally posted.
What is truth?
This hides a bigger problem. Politifact in particular and fact checkers in general do not measure truth. They measure factual inaccuracy. Some factual inaccuracy may be lies. That's essentially the purpose of the Pants on Fire rating. That's when a politician says something knowing that it is not true.
Most factual inaccuracy is just that though. Someone says something that is an exaggeration or misleadingly stated. For example, this question title currently says: "Did Donald Trump only tell the truth in 22% of his fact-checked statements?" That is a misleading restatement of "only 22% were rated 'true'. (78% were rated 'mostly false' or worse.)."
Mostly False used to be called Barely True. Such statements contain a kernel of truth. From the definition of Mostly False:
The statement contains an element of truth but ignores critical facts that would give a different impression.
So Trump told some truth in every Mostly False claim. If he hadn't, it would have been ruled False or Pants on Fire instead. Adding Mostly False to the true bin would increase the statistic to 47%.
The way to avoid Mostly False ratings is to be careful in what one says. For example:
Says "no, you’re wrong," responding to the assertion that stop-and-frisk was ruled unconstitutional in New York.
But Politifact says:
Trump has a point that a judge’s ruling in a 2013 case did not declare stop-and-frisk as a general practice unconstitutional in New York. Stop-and-frisk is still legal and still takes place in New York and across the country.
They don't claim that he wasn't truthfully describing the constitutionality of stop-and-frisk. They claim that there is an alternative interpretation of his statement (that he may not have intended) that is false:
But it also isn’t quite correct for Trump to call Holt’s claim "wrong" because that implies there was no finding of unconstitutionality in New York’s practices or that Scheindlin’s ruling was tossed out, when really it still stands.
So if Trump had said what they said, e.g.
Constitutional stop-and-frisk is legal across America
He would have gotten a True rating for essentially the same point. Fact checks frequently turn on small semantic differences like this. Note that I found this one by picking the first national claim on Trump's Mostly False page (at this moment; likely to change in the future). That's why Mostly True, Half True, and Mostly False exist: to measure small semantic differences between claims.
US News and World Report says Politifact is biased against Republicans. As do others. And others. Or see an analysis of Democrats and Republicans with different ratings for similar statements.