The Amendment reads as follows, (thanks to Think Progress for the full text of the legislation)
Though this may read like any other religious exemption clause, pay close attention to section 3. This one clause allows religious organizations and individuals to not even recognize the fact that a Rhode Island couple is in a civil union, if by doing so, they would violate their religious beliefs. This could cause problems, for example, in religiously affiliated hospitals, where a nurse who has an objection to civil unions could refuse to let someone in a civil union see his partner in the hospital.
Religious exemptions, though not necessary per the First Amendment, are an important part of any marriage equality/civil union bill. But such exemptions should not be supported when they cross the line and allow individuals to not even recognize a civil union based upon the vague concept of "religious belief". This bill, with its attached amendment, was approved by the Rhode Island legislature, and now makes its way to Governor Lincoln Chafee's desk. If he signs it, which he has said he will, the state of Rhode Island will be giving an individuals religious beliefs preference over a civil contract, creating a very dangerous precedent.