UAE nationals are exempt from doing the COVID‑19 PCR test prior to departing from the country of origin to Dubai. They will be tested on arrival in Dubai.
Children under the age of 12 and passengers with a severe or moderate disability are exempt from the PCR test.
Traveling to Abu Dhabi:
Depending on flight origin, test must be carried out between 48, 72 or 96 hours ahead of flight departure time. Travelers to Abu Dhabi from the 'green' list of countries, regions and territories will be required to present a medical certificate with a negative COVID-19 PCR test taken within 96 hours prior to their departure.
Traveling from Dubai to Abu Dhabi:
Tourists arriving to Abu Dhabi by land via other emirates, from countries included on the 'green' list will be allowed entry to Abu Dhabi after presenting a negative PCR test, valid for 48 hours.
Visitors coming in from other destinations who wish to enter Abu Dhabi through neighboring emirates will need to present a negative COVID-19 test result within 48 hours of receiving the examination, in addition to undergoing a quarantine period in Abu Dhabi if they have been in the UAE for less than 10 days (quarantine period requirement depends on the number of days spent in the UAE prior to entry into Abu Dhabi).
Children under the age of 2 are exempt from presenting a COVID-19 test.
Travelers with proof of recovery from the virus within the last 90 days prior to boarding. Proof of a recent positive viral test and a letter from your healthcare provider or a public health official stating that you were cleared to travel.
More info at the CDC site
Travelers who did not get tested, are advised to stay home and self-quarantine for 10 days after travel.