What has God to do with "Porphyria's Lover"? What is the meaning of the last line, "And yet God has not said a word!"

This is an excellent and intriguing question! Well done
for asking it! It is important to realise that Browning in this poem creates an
unreliable narrator whose words we clearly come to doubt as the poem progresses. Note
how he convinces himself that Porphyria did not suffer when he killed her, though we,
his audience, are far from certain:

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No pain felt she;

I
am quite sure she felt no
pain.

The repetition seems to
underlie the sense of doubt that we have in the narrator's account. As the poem ends, he
seems to be looking for some kind of ultimate justification that he has done the right
thing in killing her and thus possessing her forever:

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Porphyria's love: she guess'd  not
how

Her darling one wish would be
heard.

And thus we sit together
now,

And all night long we have not
stirr'd,

And yet God has not said a
word!

In the same way that
the narrator convinces himself that Porphyria felt no pain as he strangled her, so he
creates a justification for what he has done. The silence of God is seen as an approving
silence at the end of the poem, one that condones the "pure" actions of the narrator in
immortalising and cementing their love.