You will have to find a private doctor (approx cost 100-150 pln) who will do such a note for you.
Well, if you are sick so that you cannot come to the classes at the university, the doctor should write a sick leave. The only exception is any planned treatment, like when you need an operation, but it's not urgent. This is not covered by the EHIC. But everything that is needed at the specific moment (so also a sick leave if the doctor decides that you should stay at home) - EHIC entitles you for that. Another thing is when you aren't sick in fact and you need a fake sick leave, no doctor will do it for free (and it's much more likely to work if you go private), but it's illegal and it's a kind of bribery in fact...
Universities don't need formal sick notes, except for exams.
Isn't a real formal sick note, commonly known as "L-4" (this was in the past the symbol of the official form for that, now it's "ZUS some letters"), issued only to the employed people? Such a sick leave must be reported by the doctor to proper institutions so that they are able to verify whether you really stay at home and not work illegally anywhere else than at your employer.
Sometimes the doctors claim that they are not obliged to issue any sick note if you are not employed (even when the school/university accepts only sick leaves issued by doctors, necessarily with two stamps: of the doctor and of the clinic), and that's true, no law obliges them to do that (but what should you do then?), however, they finally do it when you ask them for that. It's from my experience.
The only time you might encounter trouble is with some very old school clinic where everything is done exactly as it was in 1950.
Hm... Basically even in case of the Polish citizens, if the computer system shows that the you not insured, but you know you are, you have a right to sign a statement where you confirm that you are insured. But then you have to explain it with your employer, who is responsible for paying for your health insurance (in case of people who don't work yet, it's the employer of any of the employed family members). Not a long time ago it was so that everyone had a special booklet ("legitymacja ubezpieczeniowa", commonly known also as "książeczka zdrowia") when the employer stamped that the employee is insured, then they were going to introduce the computer system, but they didn't manage to do it on time, so you could use either your old "książeczka zdrowia" (but they weren't issued any more), or you had to show a document from your employer confirming that they transferred the money for your health insurance to the ZUS (ZUS RMUA form), but now everything is finally computerized and you basically don't need any document other than your identity card or passport.
Sometimes you may have to quarrel with the ladies at the registration counter, you may have to call the NFZ (the phone numbers are here:
nfz.gov.pl/kontakt/infolinie-nfz/ - the middle column in the row dependent on the province, but don't expect them to speak English; you may also have to wait some time to get the connection with a consultant), but I anyway believe that the people in the "przychodnia" will know what EHIC is.
Sometimes it may happen that they will have no more time that day to arrange a visit at the doctor, then you may try to call other "przychodnia" and ask there. But anyway someone speaking Polish should do it.
There is also a guide in English on the NFZ website:
if you select your province, you will get phone numbers to the inernational department of NFZ, the proper phone number will be described as "Informacje dla uprawnionych z innych państw członkowskich UE/EFTA" ("Information for the entitled from the other EU/EFTA member states") - if this is a phone number for the foreigners, it's more likely that they will speak English there.
you basically don't need any document other than your identity card or passport
For clearity, it's about Polish citizens, of course - they don't have foreigners in their database and you need EHIC if you come from abroad.