I'm not overly familiar with the Constitution. Does it actually say that?
No, it doesn't say that; but some believe (based on the other writings of the authors, and of the amendments), something to this effect. But it's not the only interpretation. I would say that everyone has the responsibility to live according to the law of the land (of which the Constitution is the highest declaration of that law). Under the Constitution and common law, individuals have rights to choose to do or not do a wide variety of things, at their pleasure; government, on the other hand, is limited in what it can make people do and how it can relate to people. The Constitution is an agreement between "We the People," intended to address certain problems of societal governance in a non-religious way.
Religion is not so based, and the Constitution specifically prevents government from having or being able to take a position on religious matters.
Anyway, I disagree with the idea that there is any implied imperative to act on the part of individuals in the Constitution. Even if we take the position that the "right" thing to do is to follow the intent, it does not, and it in fact would be contrary to the spirit of the Constitution to REQUIRE certain behaviors. For example, the Constitution recognizes the right to vote; but it is up to individual discretion whether or not to vote--requiring all qualifying adults to vote is contrary to the freedom of the individual to choose how to engage in the democratic process as laid out in the Constitution.