Many jurisdictions recognize weddings performed out-of-jurisdiction provided that those weddings are legal in the jurisdiction they're performed in. Unfortunately, there's no legal authority with jurisdiction over LEO -- it's kind of a legal gray area.
Does any national law system provide a means for marriage officiants to lawfully perform a marriage at an arbitrary location outside of the territorial jurisdiction of the authorizing state (which could include space, even if not explicitly mentioned in the law)?
Marriage by proxy, as in the Malenchenko case mentioned by @PearsonArtPhoto, is legal in some jurisdictions:
In the United States, proxy marriages are provided for in law or by customary practice in Texas, Colorado, Kansas, and Montana Of these, Montana is the only state that allows double-proxy marriage. Proxy marriages cannot be solemnized in all other U.S. states.
...but as @called2voyage points out, the marriage is performed on Earth.
There's a tradition of allowing ship captains to perform marriages at sea in international waters; those marriages are often but not always recognized:
In one well-known case, Fisher vs. Fisher, a court ruled that a particular marriage solemnized by a ship's captain was valid (and more generally that, absent a statute stating otherwise, an exchange of vows between two consenting parties constituted a valid marriage). In another case, Norman vs. Norman, a court came down on the opposite side of the fence.
Fisher vs Fisher was a 1929 case in New York state; Norman vs Thomson (frequently cited as Norman vs Norman) was an 1898 California case, but the wedding was invalidated because it was held that the couple married at sea…
for the avowed purpose of evading the statute of the state requiring a license and solemnization by an authorized person.
Given those precedents, I suspect that if the commander of the ISS performed a wedding of a couple of New York residents aboard, and it appeared that the marriage was entered into in good faith, that New York state would recognize the marriage upon their return home, and that the marriage would be thus recognized throughout the US via the full faith and credit clause of the Constitution.