Religion actually leads to independence in different ways for both the characters Nora in A Doll's House and Juliet in Romeo and Juliet. For Nora, the denunciation of religion actually leads to her independence, while for Juliet both the embracement of her religion and denunciation leads to her independence.
In Henrik Ibsen's A Doll House, religion is not directly mentioned until the final scene when Torvald accuses Nora of having "no religion, no morality, no sense of duty" and accuses her father of raising her with "want of principle." Later in this scene, Nora even agrees with him, saying,
I am afraid, Torvald, I do not exactly know what religion is...I know nothing but what the clergyman said, when I went to be confirmed...When I am away from all this, and am alone, I will look into that matter too. I will see if what the clergyman said is true, or at all events if it is true for me. (Act 3)
Through this passage, we learn that in feeling subjected to the duties of a wife as laid out by society and the church, Nora feels that she does not only want to abandon her husband, she wants to abandon all things that teach such subjection, even religion. Since religion teaches that her duties as a wife and mother come first, her abandonment of religion helps to lead Nora to her independence.
Juliet, however, relies on religion to guide her. In accordance to religion, she refuses let Romeo have sexual relations with her before marriage, saying, "What satisfaction canst thou have to-night?" and then further asks him,
If that thy bent of love be honourable,
Thy purpose marriage, send me word to-morrow. (II.ii)
This passage testifies that Juliet is bent on conducting their relationship in accordance with her religion; she will only let Romeo near her if he marries her. Since Juliet does not ask her parents' permission before agreeing to marry Romeo, we see that unlike Nora, Juliet gains independence through following her religion. Juliet even follows her religion in not asked her parents' permission, because parental consent at her age was not demanded by the church.
However, one way in which she breaks away from the creeds taught by her religion is in her determination to commit suicide. When she goes to Friar Laurence for counsel on how to break her betrothal to Paris, she threatens that if he cannot think of a solution, then her own solution will be to kill herself with a knife. We see this in Juliet's lines, "Do but call my resolution wise, / And with this knife I'll help it presently" (IV.i). Since taking one's own life is considered one of the gravest sins in Christianity, in wanting to commit suicide, Juliet is breaking away from her religion. However, her choice to break from religion leads to her independence because her willingness to kill herself gives Friar Laurence the idea to help Juliet fake her own death in order to be with Romeo.
Hence for Nora, breaking away from her religion leads to independence, while for Juliet, both remaining faithful to it and breaking from it leads to different moments of independence.