Listen to last night's class, "R. Ovadia Yosef & Secular Jews," here.
Follow along with the sources here.
For further research:
1) Listen to our halakhah class, "The Status of Today's Mehalelei Shabbat" here.
2) Read Adam Ferziger's analysis of Binyan Sion's teshuvah (Source 1) and several other related decisions of R. Yaakov Ettlinger, in chapter 4 of Exclusion and Hierarchy, here.