Squeezing spermicide out of a tube is permitted on Shabbat.The question is spreading it. If the spermicide has a more liquid consistency (i.e. it does not hold its shape when squeezed out), then it is permitted to spread it as needed on Shabbat. But if it is more of a gel or cream, then spreading it would be a form of prohibited labor.
One solution to prevent having to spread a gel or cream on Shabbat would be to apply the spermicide to the edges of the diaphragm before Shabbat. You could then squeeze some into the middle on Shabbat (so that it spreads itself), just prior to insertion.
An alternative might be to insert the diaphragm shortly before Shabbat. The diaphragm and spermicide may be safely and effectively inserted up to six hours prior to relations. You can verify the exact parameters by looking at the instructional inserts that come with your diaphragm and spermicide. You could also top it off with spermicide from the applicator on Shabbat.
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This response was updated on 16 December 2020.