Reference instructions for religious works were only recently added to the APA Publication Manual in the seventh edition.
Here are the rules collected from the APA Publication Manual Seventh Edition (examples follow).
Revised Standard Version Catholic Edition (RSVCE) from an online source and an annotated edition:
Revised Standard Version Catholic Edition. (1966). Revised Standard Version Catholic Edition (RSVCE) - Bible Gateway. https://www.biblegateway.com/versions/Revised-Standard-Version-Catholic-Edition-RSVCE-Bible/ (Original work published 1611)
Kaiser, W. C., Jr., & Garrett, D. (Eds.). (2006). NIV archeological study bible: An illustrated walk through biblical history and culture. Zondervan.
Citation in parentheses:
(Revised Standard Version Catholic Edition, 1611/1966)
(Revised Standard Version Catholic Edition, 1611/1966, Song of Solomon 8:6)
(Kaiser & Garrett, 2006)
Narrative (title or author is used in a sentence):
Revised Standard Version Catholic Edition (1611/1966)
Revised Standard Version Catholic Edition (1611/1966, Song of Solomon 8:6)
Kaiser and Garrett (2006)
Instructions and examples taken from: https://apastyle.apa.org/style-grammar-guidelines/references/examples/religious-work-references and sections of 8.13, 8.28, 9.42, and 10.2 of Publication Manual of the American Psychology Association, 7th ed., 2019.
The APA Publication Manual doesn't explicitly contain instructions for Church or Papal documents. Here are suggestions from the APA blog (according to APA 6).
Laudato Si' using both online Vatican web source and print book from Melville House.
Pope Francis. (2015). Laudato Si’ [On Care for Our Common Home] [Encyclical letter]. Retrieved from: http://www.vatican.va/content/francesco/en/encyclicals/documents/papa-francesco_20150524_enciclica-laudato-si.html
Pope Francis. (2015). Encyclical on climate change and inequality: On care for our common home. Melville House
Citation in parentheses:
(Pope Francis, 2015)
Narrative:
Francis (2015) declared...
This information is taken from this APA blog post: https://blog.apastyle.org/apastyle/2017/05/whats-in-a-name-names-with-titles-in-them.html. The instructions are for APA 6 however there's been no substantial change in this part of the APA manual between versions 6 and 7.