Apacite displays month in references

You could try using the apalike style instead:

\documentclass{article}
\usepackage{natbib}
\usepackage{filecontents}

\begin{filecontents}{tbib.bib}
@article{fulton_religious_1999,
    title = {Religious orientation, antihomosexual sentiment, and fundamentalism among Christians},
    volume = {38},
    issn = {0021-8294},
    doi = {10.2307/1387580},
    number = {1},
    journal = {Journal for the Scientific Study of Religion},
    author = {Fulton, Aubyn S. and Gorsuch, Richard L. and Maynard, Elizabeth A.},
    month = mar,
    year = {1999},
    keywords = {Homosexuality {(Attitudes} Toward), Religious Beliefs, Religious Fundamentalism, religious orientation \& Fundamentalism \& antihomosexual sentiment, 18–24 yr olds},
    pages = {14--35},
}
\end{filecontents}
\begin{document}

\cite{fulton_religious_1999}
\bibliographystyle{apalike}
\bibliography{tbib}

\end{document}

I also find this very frustrating. Including the following line somewhere after beginning the document solves the issue in case you never need the month included.

\renewcommand{\APACrefYearMonthDay}[3]{\APACrefYear{#1}}