Yes. Although some states have a waiting period after you get the license and some states don't. In New York, it is 24 hours; In Nevada, it is none.
Why are there two steps? When they are approved to get a license, why does the couple then have to go through another procedure to conclude the process? Is it just custom? Aren't the two signatures on (what is now) the license enough for the government to recognize their new legal status?
Technically, the marriage license is the second, added step. The wedding ceremony predates it by ... all of human history.
As far as the civil government it concerned, the reason for a wedding celebration is to formally impress upon the bride and groom the gravity of the commitment they are making. In fact, that is the only requirement. There are no particularly required words to say or rituals to perform. Any "sufficiently serious" ceremony is acceptable.
Merely signing a piece of paper is not considered by the government to be sufficient evidence that one understands and consents to entering into the status of matrimony.
Perhaps you find this illogical. Perhaps you think there is no reason for the law to operate this way. That does not change the fact that the law exists, that this is the reasoning behind it and that it will never, ever change.