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But in this particular instance, there is nobody's due process right being violated by a state judge saying abortion is legal.

What about the people of MN and their state legislators?

The people of MN elected their state legislators. Their state legislators man anti abortion laws.

If the MN ruling arbitrarily strikes down these laws (I've not read the MN Constitution, nor the laws), then the due process rights of the people of MN are being abridged.

There's no such thing as "general citizen" standing, so you'd never even get to the merits.

Making due process claims possible when there is no substantive right being violated would create a federal case out of every single state case where one party believes the court "got it wrong". I think the federal courts could and wood rule that they have no jurisdictions over the general question of whether or not state law has been correctly applied in a given case. They've never done it before, and it would effectively make every single state case appealable to SCOTUS.