Whether or not one agrees with the interpretation, there is a reasonable biblical case to be made that a woman should not preach or teach in the church. Paul writes in 1 Tim 2:
8 I desire then that in every place the men should pray, lifting holy hands without anger or quarreling; 9 likewise also that women should adorn themselves in respectable apparel, with modesty and self-control, not with braided hair and gold or pearls or costly attire, 10 but with what is proper for women who profess godliness—with good works. 11 Let a woman learn quietly with all submissiveness. 12 I do not permit a woman to teach or to exercise authority over a man; rather, she is to remain quiet. 13 For Adam was formed first, then Eve; 14 and Adam was not deceived, but the woman was deceived and became a transgressor. 15 Yet she will be saved through childbearing—if they continue in faith and love and holiness, with self-control.
A theological justification for the point then is this - Eve was deceived, and therefore all women can no longer be trusted. It is, in fact, Scriptural, using a very plain sense reading of the text.
That said, there are varying interpretations and contra-indications to such a plain instruction. (2 John being addressed to "the elect lady and her church" comes to mind.) But, if the standard interpretation is the sensus plenior here, then the Bishop of Rome would simply being saying that the church has no authority to create a new teaching out of a very old Scripture.