The 36-year-old actor-and-comedian - who tied the knot with the 'One That Got Away' hitmaker in India in October 2010 - explained despite the pair allegedly parting ways, he is very "happily married" and will remain that way "until death do us part".
He told TV host Ellen Degeneres: " There are always rumours, aren't there? In the end you have to just not engage with internet technology. I've treated the whole internet now like it's a wicked little liar. I am really happily married.
"Being famous is like a little bit of you is taken away and goes off and lives on it's own and does what it wants. I wish it would do more interesting things. I don't know what he's doing, the other me, but I'm married to Katy. Perpetually, until death do us part was the pledge - and I'm still alive."
Russell also admitted his paternal instincts have kicked in since he was asked to be godfather to his friend's one-year-old twins.
When asked if he and Katy are planning a family, Russell replied: "I would like one. I love those little babies in the beginning. They're so lovely in that bit. Me and Katy are godparents to two twin boys. They're called Sam and Donovan.
"At first they were boring. They were just essentially little fluid factories, generating fluids - no regard for what I was wearing. Now, they are interesting. They smile and look at me and wonder - I'm just observing them like I'm a scientist."