The passage in question reads, "So she said, 'See, your sister-in-law has gone back to her people and to her gods; return after your sister-in-law.'" To put the story in perspective, Naomi's husband and sons find themselves in Moab, southeast of the Dead Sea, because there was a famine in Bethlehem. Naomi's husband dies and her sons marry Moabite women. By and by both sons die and Naomi decides to return to Bethlehem. However, her daughter-in-laws decide to go with her. Naomi convinces Orpah to go back, but Ruth insists on coming along. This is where we find the passage in question. And it is where we find one of the most popular wedding scriptures, "Where you go, I will go; where you live, I will live; your people shall be my people, and your god shall be my God" (1.16).
But is Naomi (and the writers of Ruth) implying it's okay for indigenous people to worship their own gods? Perhaps. And perhaps not. There are a number of possibilities why this sentence may have been included.
The first possibility is this could be a historical story and this is what Naomi actually said. But the question follows, why would she spout such heresy? There are two explanations. For one, there was a belief in early Hebrew history where gods connected specifically with a geographic location; therefore, it was normal for an individual to worship the gods associated with the locale. Some believe the reason Israelites built altars and shrines at the sites of epiphanies (revelations from God) was because the God of that locale was somehow restricted to that specific location. This is one of the reasons why the Israelites worshiped the Canaanite gods in the Promised Land. There was an assumption the Canaanite gods were residents/controllers of the land. Therefore, if Ruth returned to Moab it would be acceptable for her to worship the gods of her people and unnatural for her to worship the God of Israel.
Another explanation why Naomi may have told Ruth to return and worship her own gods is because the Israelite faith assumed the woman's faith was subservient to her husband's or to her father's faith. Indeed, in Israel a woman's faith was considered so inferior to a man's that they couldn't even take a vow without the approval of their husband or father (Numbers 30). Since Naomi was trying to send Ruth back to her own people, there was an assumption she would remarry, thus she would have assumed Ruth would re-adopt the faith of her new family.
The problem with these two possibilities is that in either case Naomi is still giving her tp worship foreign gods. This doesn't seem to be typical (or even acceptable) to what we consider an Israelite norm.
But there is a second possibility, considered by many to be a more compelling explanation. Most scholars believe the book of Ruth was written sometime after the Babylonian exile as a rebuttal to Ezra's command of divorce. During the seventy year Babylonian exile the men left behind married women from Moab, Egypt, Canaan, and so on. Ezra wanted these men to divorce these "impure women" and send them and their children away (Ezra 9-10). So the book of Ruth may have been written to suggest there were already foreign women in the lineage of the Israelites, including the great King David (Ruth 4.13-17) and to repudiate Ezra's command.
If this is the case, and scholars are generally united that it is, then the words of Naomi may be a poignant slap against the religious leaders who wanted to dismantle these families. Her message was, "If you send back those women to the lands of their ancestors, then they will no longer be worshiping the God of Israel." Thus the religious leaders would be guilty of approving and facilitating idolatry -- not a charge these leaders wanted to have.
So is Naomi really approving of the worship of other gods? Perhaps, but perhaps not. It depends on your perspective.