To say that a marriage is invalid is to say that there was an obstacle (technically, an impediment) which prevents the couple from actually being married (not just from "being married in the eyes of the Church"). Some of these obstacles are matters of divine law—for example, canon 1091 section 1:
In the direct line of consanguinity marriage is invalid between all ancestors and descendants, both legitimate and natural.
This is a statement given by God (Deuteronomy 23:1). As such, it is part of the natural law, binding on all people.
There are other impediments which are purely ecclesiastical in nature. In other words, they were created specifically by the Church, not by God. For example, canon 1083 section 1:
A man before he has completed his sixteenth year of age and a woman before she has completed her fourteenth year of age cannot enter into a valid marriage.
God never specified exact ages before which a man or a woman couldn't marry (although it is clear that one can't get married at an arbitrarily young age). But the Church has decided that sixteen and fourteen years are reasonable age limits, and legislated accordingly.
Impediments which are matters of divine law are applicable to all people, whether Catholic or not—a marriage between a non-Christian man and his mother would still be invalid. But what about impediments that are ecclesiastical?
It seems unfair to apply rules that were specifically created by the Catholic Church to those outside the Church; and indeed, though the impediments of divine origin are binding on those outside the Church, they are so not because they are canon law, but because they are natural law. The Church just has specific implementations of them in canon law.
For a couple outside the Church, then, as long as a marriage is performed in accordance with natural law, and with the law of the land, it is a naturally valid marriage. In this case it is possible that the couple are subject to some ecclesiastical impediment, but still naturally validly married.
In the case of a marriage involving at least one Catholic, however, the case is different. God has given the Church authority to make these laws and to bind them on all Catholics. Thus, if the couple have ignored an impediment listed in canon law, their marriages are in fact invalid.
On the other hand, as canon law (1060) states:
Marriage possesses the favor of law; therefore, in a case of doubt, the validity of a marriage must be upheld until the contrary is proven.
In other words, if no one is asked whether a given marriage is invalid, or if an investigation is ongoing into its validity, it is presumed valid unless proved otherwise. This means that there may be some marriages which are presumed valid but are in fact invalid. The reverse, however, is not the case, at least for a Catholic marriage: there are no marriages which are in fact valid ("in God's eyes") though found invalid (by a marriage tribunal).
To address a couple of comments: If the diocesan marriage tribunal determines that a marriage is invalid, but the couple does not accept the decree of nullity, the marriage is still invalid, as stated above. And (again as stated above) if the validity of the marriage is never investigated—if the couple does not seek an annulment—it is still possible for the marriage to be invalid: but the Church is under the obligation to treat it as valid.