heart monitor cat5 cable on back home internet

If the heart monitor is designed to be connected via Ethernet and communicates/reports to a monitoring application on a client on the network it should be okay. I suggest checking the manual/information for the heart monitor to be sure. There are Ethernet ports on the gateway and a wired client would obtain an IP address via DHCP if it is configured to do so. Check the documentation for the heart monitor and then you will have a clearer answer. If you don't have a manual for it then search for the model of the heart monitor online for documentation.