Short answer: he did not.
Neither did Prince Myshkin, that we know of.
Likely both believed it. Beauty — in the person of Christ — will do so.
And clearly D wrote of M in The Idiot to explore art and beauty and ugliness and salvation.
But did he say it, and did he believe that a person could live so as to embody it — to re-incarnate Christ in our lives.
He did not.
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Some discussion of it here.
And herein my longer treatment of it, tho not long enough.
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For this we need life eternal, and the Life.
Image:
The Body of the Dead Christ in the Tomb [detail]
Hans Holbein the Younger, via the Public Domain
