Thanks @cole! Your suggestion gave me a much more helpful error message.
con <- curl::curl(url = "https://myserver.com:port")
readLines(con = con)
# Error in readLines(con = con) :
# Peer's Certificate issuer is not recognized.
RStudio Server Pro and RSConnect are installed on the same virtual machine and are listening on different ports. There is no proxy server. The SSL certificate is was issued and signed by our local certificate authority. The local CA's root certificate is installed on all company computers but might not be installed on the Linux VM where the RStudio software is running. I don't think we would consider this a self signed certificate since it was signed by a CA (internal) with a root certificate, correct?
So I think my next steps will be to add the local CA's root certificate to the list of trusted CA on the Linux VM by following the instructions in your first link.