Yes. Not by the magic of the fact that it is in writing, but by the fact that it has helped immensely in establishing and propagating conservative versions of the language, based on written records, as the most prestigious versions, which are learned in education and emulated in formal registers.
Given the time depth of the Mayflower, American English should really be a separate language from England English by now. And true enough, there are issues with mutual intelligibility in some registers. But the written norms of the two are close enough, and universally propagated enough, to have kept them in sync.
Universal literacy, and familiarity with the sagas, is widely held as the reason Icelandic has changed relatively little in the past 1000 years. It’s also one of the few places where prescriptive intervention has actually reversed a language change (flæmeli). Written Greek has had a similar effect on Greek dialect.