We really don't use the terms "ruling party" or "opposition party" in the US.
The 2 party system in the US is an incidental system - not one by constitutional design. It just happened that 2 parties have most of the votes in government, but other parties exist, and in fact, some independent candidates run, and sometimes win.
The Reform party and Green party are examples of small political parties that field candidates in the US. Senator Joe Lieberman of Connecticut and Senator Lisa Murkowski of Alaska are examples of independents (not belonging to a political party).
As you can see, it is not truly a 2 party system either.
In our system, the president is visible, but doesn't really have any power in the legislative process. He also has no power in the selection of legislators. His party could not really be called a ruling party. (Americans have a particular distaste for terms like "ruling party" even if it fit our system.)
In each house of congress, since the Democratic and Republican parties make up the overwhelming majority, we tend to use the terms "Majority party" and "Minority party" depending on which of those parties have that position. Currently in the US, the Republican party is the majority party in the House of Representatives, and the Democratic Party is the majority party in the Senate. This is another reason that "Ruling party" and "Opposition party" does not fit our system and isn't used.
But to your question, I will agree with greensilence that if a coalition is formed as often happens, then you might call that the ruling coalition if it really is a ruling coalition. In a system like the US system, I doubt such a coalition could be formed. If a ruling party or ruling coalition exists, and there are parties that do not join or fit with that coalition, then they could be called "opposition parties."