The title says it all. I've tried digging up debate on this issue to see a proof one way or the other but it doesn't look like anyone is able to say whether or not it is. Clearly there are recursive structures in English, and that's about all anyone seems to say.

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    $\begingroup$ Do you have any formal definition of (written) English in mind? English is a constantly changing object that means different things to different people, and has no standard definition. $\endgroup$ Jan 8, 2016 at 9:10
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    $\begingroup$ If you restrict the length of sentences to, say, 4000 words, then English is even regular (since you have finitely many words, whatever dictionary you use, and can only produce sentences of finite length, giving finitely many sentences). If you allow arbitrarily long sentences, then just saying "English" doesn't help all that much since you would need a formal grammar now to answer the question. $\endgroup$
    – G. Bach
    Jan 8, 2016 at 10:00
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    $\begingroup$ @G.Bach Yes, I'm reasonably sure that all natural languages are regular, because our brains can only hold a finite amount of state and can only do a finite amount of computation (eg use external storage) before we die. As natural languages are defined by their use by humans, constructs that no human can produce or understand are not part of the language. $\endgroup$
    – adrianN
    Jan 8, 2016 at 11:01
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    $\begingroup$ @adrianN - It is actually known that Swiss-German is not context free. See here. $\endgroup$
    – Shaull
    Jan 8, 2016 at 11:29
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    $\begingroup$ As far as I know, Chomsky himself stated that even his type-0 model did not capture natural language. So the answer would be: no. $\endgroup$
    – Raphael
    Jan 8, 2016 at 11:47