Juliet swearing by Saint Peter can be seen as blasphemous because, during the Elizabethan era, one of the tenets of the Anglican church held that it was blasphemy to worship any saint or to invoke the intervention of any saint in human affairs.

In act 3, scene 5 of Romeo and Juliet, shortly after Romeo leaves Juliet to travel to Mantua early in the morning after their wedding night, Lady Capulet comes to Juliet’s room to tell her about the plans that Lord Capulet made with Paris for Juliet to marry Paris in just a few days.

LADY CAPULET: Marry, my child, early next Thursday morn

The gallant, young, and noble gentleman,

The County Paris, at Saint Peter's Church,

Shall happily make thee there a joyful bride.

JULIET: Now by Saint Peter's Church, and Peter too,

He shall not make me there a joyful bride!

Here, Lady Capulet tells Juliet that she’s to be married to Paris in Saint Peter’s Church in Verona. Juliet expresses her displeasure with the sudden, unwelcome marriage arrangements by invoking "Saint Peter’s Church," which is to say, the original Roman Catholic Church (not simply Saint Peter’s Church in Verona), and to invoke the intervention of Saint Peter himself to support her decision not to marry Paris.

Lady Capulet is taken aback by Juliet’s response to the news of the arrangement—"I would the fool were married to her grave!" (3.5.143)—but it is Juliet’s father, Lord Capulet, who threatens to "drag thee on a hurdle thither (to Saint Peter’s Church)," and he further threatens to disown and disinherit Juliet if she refuses to marry Paris.

LORD CAPULET: An you be mine, I'll give you to my friend;

An you be not, hang, beg, starve, die in the streets,

For, by my soul, I'll ne'er acknowledge thee,

Nor what is mine shall never do thee good.
(act 3, scene 5, lines 200–203)