A lone observant Jew is stranded on a small pacific island. Under what circumstances may he take one of the island women as a wife and procreate from her?
The elements: Must the Jew remain celibate? Can he perform a conversion of any standing?
He may take a local woman as his wife in the circumstance where he has two Jewish friends along with him (who are not related to him). There are historical cases where Jews traveled to far-away lands and settled there, taking wives from the local population. ( Conversion for the sake of marriage is permitted according to Yevamot 24b. )
I don't think it would be possible to halachically marry someone under the conditions in the question. The woman would need to convert (Bereishis Rabbah 65), which requires a beit din of 3 Jews (Yevamot 46b), and the marriage would require a ketubah and two kosher witnesses and all the other accouterments of a halachic marriage (see Tractate Kiddushin, Rambam Mishne Torah Hilchot Ishut).
However, if somehow one claimed to have done so, it may be considered as a valid marriage, and he may be required to give a get if they wished to divorce. (See this article for an in depth analysis.)
A similar situation is dealt with here.
Must the Jew remain celibate?
This is not the same as asking whether he may take one of the women to be his wife. It may be possible to somehow have relations with her if he takes her to be a concubine (Sanhedrin 21a, Rama on Shulchan Aruch EH 26:1), slave (shifcha), or as an eishet yifat toar. However, this is an extremely complicated topic, and beyond the scope of this question. Intercourse with non-Jews is prohibited (Avoda Zara 36b, Shulchan Aruch EH 16:1).
Can he perform a conversion of any standing?
No, for conversion requires a beit din of three Jews. (Yevamot 46b)