No, federal law only prohibits the federal government from recognizing same-sex marriage. It does not prohibit same-sex couples from marrying.
It shouldn't. Ask about it.
Although the US federal government began recognizing same-sex marriages on June 26, 2013, it is state law that controls who can and cannot be married. Therefore, each state may choose to legalize same-sex marriage or prohibit it independently. Experts predict, however, that same-sex marriage will be legal in all fifty US states by 2018.
A federalist would say that the US Federal Government should either force states to permit same-sex couples to marry (e.g., by order of the US Supreme Court), or prohibit them from doing so (e.g., by amendment to the US Constitution).
strict interpretation
Yes
You should approach the US embassy for accurate information or just register your marriage with the Cambodia government. The marriage will be recognized by the US government.
through the naturalization process required by government
The US Constitution does not explicitly speak of the requirements for obtaining a marriage license, nor of the legal recognition of marriages. This aspect of family law has historically been left up to the individual states.
You better register the marriage at the nearest US mission in the country where you are getting married to avoid the risk that the US government may not recognize marriage certificate from that specific country.
In general, a marriage is recognized by the federal government and by U.S. state governments if the marriage was considered legal in the jurisdiction in which it was performed (for example, Canada), unless the marriage is against public policy (i.e., incestuous, polygamous, etc.). One important exception is same-sex marriage, which is currently not recognized by the U.S. federal government and by many state governments. States that do recognize same-sex marriages, however, will also recognize a same-sex marriage performed in Canada.
In the United States, the states have always regulated eligibility and validity of marriage. Only once before, in the issue of interracial marriage, did the federal government overrule states rights in this regard.
A Justice of the Peace in Louisiana has been turning down interracial couples that want to get married. You can be pretty sure that even after same-sex civil marriage is legal all over the United States, there will be Justices of the Peace who will turn them down, too. Some people will never learn! Another problem is confusion between civil marriage and religious marriage. Some religions will probably never allow religious marriage. But in the US, they can't make the government prohibit civil marriage.