As a matter of chronology, yes, the United States was the first major Christian country (and it was fairly substantial, even at the time of the Revolution) to not require the establishment of a particular state church. Not all colonies had established churches (Rhode Island, since 1636!, Pennsylvania, Connecticut, etc...) but most colonies disestablished in the 1780s. Virginia, for example, disestablished her Anglican church in 1786. Congress was prohibited from establishing a federal church in 1791 -the same year that France disestablished hers in the midst of the French Revolution.
Different from establishment, however, is the notion of separation of church and state. The Constitution of the United States bars the U.S. Congress from establishing a state church (Amendment I), but the "wall of separation" developed slowly over the next few centuries, and was not formalized until Lemon v. Kurtz in 1968. Whether and how this "wall" is supposed to work is an ongoing discussion.
The Netherlands (1795) and Switzerland (1798) disestablished their churches about the same times, but by a strict chronology, the United States was absolutely first. Germany (or more properly the Holy Roman Empire), left it to each prince to decide the status of their particular states prior to all of these, but the establishment of a church was considered important. This chart on Wikipedia gives many more dates.
The problem with disestablishing a church is not as straight-forward as it might seem, however. In Virginia, for example, the Anglican church was the governmental authority charged with looking after the citizenry. If a person was unable to afford food, for example, it was the established church that cared for that person, using taxes levied and collected by the church. Until a parallel, secular authority could be instituted to serve the same role, disestablishment would have hit the "least of these my brethren" extremely hard. Indeed, even the Baptists who hated the established church so much were required to show that they could care for their members before the Act of Toleration in 1786.
Update: Per this article (link) Massachusetts did not disestablish her church until 1833, but as a Virginian, it is very hard to consider Massachusetts a major state (joke). In all seriousness, that article shows how long and hard the process of disestablishment was - but at the federal level, it was much easier, because technically, it never could begin!