I think your starting point is wrong: you design a (medical/medical-related?) device featuring a communication (Ethernet/other) port, not an Ethernet (what?) featuring a device. Technically, from the standardization point of view, there is a sort (category) of electrical/electronic equipment having (among others) a communication ability.
First, you need to determine as tighter as possible the category your future device falls in, then, second, search for the requirements (common and/or particular) applicable/mandatory for the category (of the equipment) in the interested jurisdiction. In Europe, you could begin from IEC.
Returning to the category, ICS Number for the group "Medical Equipment" is 11.040, you can/need to certain the number deeper depending on your design, to find out the standards related to it. Try, for example, here.
After you (yourself, or consulting a specialist in that area, such as in Kema or other certification/test body) find the applicable requirements on (the category of) your equipment, such a standard very probably would be an umbrella one, i.e. it will contain references to (the requirements of) many/some other standards.
For example, it (very probably) will have (among others) a reference to EMC-related standards, including both susceptibility and emission (CISPR); EMC-related standards (typically) know nothing about Ehternet, but much about communication ports. Taste the IEC 60601 series (yes, each IEC/ISO/IEEE standard costs perceptibly).
If my device passes EMC tests,
Which tests? What are the requirements?
and my Ethernet magnetics pass isolation tests,
Nobody cares, what about the device in the scope of the requirements?
and my Ethernet port works with a crappy 100 m long cable,
Nobody cares again before the requirements on the devices are clear, for particular Ethernet PHY there is the corresponding PICS, are your implementation satisfying it?
is there some sort of Ethernet police which can come and get me, if I say that my device has Ethernet port, but I did not do the Ethernet compliance tests?
Typically and probably out of the context of a medical-related device, such a compliance could be declarative and not necessary approved by a specialized testing/certification body, well... such a police could be an angry client/community spreading out its "grounded, bad user experience" relating to your device, in a social network or forum in the internet - any is well indexed by Google and other search engines.