Suppose a Jewish man and Jewish woman end up stranded on an island in the South Pacific as a result of a plane crash or ship wreck, with no other survivors, and no hope of rescue. In fact, respective families had already said Kaddish for them.
As a result of struggling to survive over the years, the two eventually fall in love.
Is there any way they can declare themselves married in God's eyes alone?
"Married in G-d's eyes" is an awfully hazy phrase. We believe that G-d gave us laws that tell us what marriage is (and isn't).
What effects marriage between a Jewish man and woman, in theory, could be relations, but that would require intent and witnesses (well witnessing seclusion). Maimonides, Laws of Husbandry Ch. 3
And the Talmud says this is a theoretical, and imposed lashes on anyone who tries this crazy stunt. Giving a ring (or anything else intrinsically worth a dollar, for that matter) before two adult, Jewish male witnesses is the way to go.
And no, "everyone knows they're shipwrecked" doesn't mean there are witnesses to seclusion, that means people assume they're dead. (If everyone on the street sees Joe & Jane living as a family unit for a year, that's a very different story.)
But without witnesses, no there is no "lifeboat option." There's a famous story of a Jewish man and woman who are taken as slaves by the Romans and put in a room together to reproduce; as there was no possibility of marriage, he didn't touch her.
We have a system of laws, and they are what they are. Trying to to somehow creatively squeeze the law into legitimizing our actions really smells of hubris. Recall how King Saul kept insisting, through all sorts of contortions and rationalizations, that he followed the law when he hadn't (I Samuel 15:20), and this caused him to lose the throne. King David, on the other hand, (II Samuel 12:13), has the courage to admit "yes I messed up." -- and he's able to survive his failure.
Regarding the shipwreck case or whatever, I can theoretically point out that premarital relations are less-seriously prohibited than most other prohibited relations (e.g. adultery bears a theoretical death penalty, whereas premarital relations are at most lashes), and I have no idea how G-d would judge particular people in particular situations. But don't try to play games saying that this is within the boundaries of the law.
(As for "what about the commandment of procreation?" The occasional situation can come up in which well, you can't always fulfil every "yes-do" commandment. If there's a mean guy who bought every Etrog in a thousand mile radius just before Sukkot and won't share them, no you are not allowed to steal one of his -- you just turn to G-d and say "I tried my best but couldn't hold an Etrog this year", and if it was truly and legitimately beyond our control, we're not held to task for it -- just as the Torah makes clear that we never blame a rape victim.)
