(with apologies to Yakko Warner)
United States, Canada, Mexico, Panama,
Haiti, Jamaica, Peru,
Republic Dominican isn't Dominica,
Cuba, El Salvador, too.
Barbados, Colombia, then Guatemala,
Honduras, Brazil, Costa Rica,
Chile, Venezuela, Guyana, Bolivia,
Ecuador and Argentina.
St. Lucia, Grenada, Belize, Nicaragua,
Bahamas and the Philippines,
Fiji and Paraguay, Suriname, Uruguay,
St. Vincent and the Grenadines.
St. Kitts and Nevis, Antigua/Barbuda,
Trinidad also Tobago,
Liechtenstein, Monaco, Denmark owns Greenland,
as well as some islands called Faroe.
Norway, and Sweden, and Iceland, and Finland,
And Switzerland, and San Marino,
France, and Bulgaria, close to Slovenia,
Italy, Greece, Montenegro.
Andorra, Romania, Poland, Albania,
Czech now Republic, UK,
The Netherlands, Luxembourg, Belgium, and Portugal,
Vatican City and Spain.
There's Macedonia, Kosovo, Serbia,
Ireland, Hungary, Russia,
Turkey, Slovakia, Germany, Austria,
some of which used to be Prussia.
Bosnia/Herzegovina, Croatia,
Tajikistan, Jordan, Ukraine,
Then South Ossetia, Qatar, Transnistria,
Lebanon, Malta, Bahrain.
Armenia, Kazakstan, Latvia, Kyrgyzstan,
Georgia and Azerbaijan,
Estonia, and Lithuania, Belarus,
Moldova, Turkmenistan.
Abkhazia, Cyprus (both Northern and otherwise),
Then Nagorno-Karabakh,
Afghanistan, Pakistan, Oman, Uzbekistan,
Egypt, Sudan, and Iraq.
Eritrea, India, Israel, Syria,
Yemen, Iran, and Kuwait,
Somaliland, Saudi Arabia, Palestine,
United Arab Emirates.
China, Malaysia, Hong Kong, and Australia,
Thailand, Nepal, and Bhutan,
Bangladesh, Singapore, Myanmar, and East Timor,
Two of Korea, Japan.
Maldives, Mongolia, Laos, Indonesia,
Taiwan, Tonga, and Tuvalu,
The microstate Sealand, Sri Lanka, New Zealand,
Cambodia, Vanuatu.
Micronesia, and Brunei, Kiribati, and Niue,
Nauru, Palau, Vietnam.
Papua New Guinea, Samoa, the Islands of
Cook, Marshall and Solomon.
Tunisia, Morocco, Uganda, Angola,
Zimbabwe, Djibouti, Botswana,
Comoros, Zambia, Swaziland, Gambia,
Guinea, Algeria, Ghana.
Burundi, Lesotho, then Malawi, Togo,
Burkina Faso, Centrafrique,
Niger, Nigeria, Chad, and Liberia,
Kenya, Benin, Mozambique.
Tanzania, Somalia, Congo-Kinshasa,
Sierra Leone, and Sahrawi,
Mauritius, and Senegal, South Sudan, Libya,
And Equatorial Guinea.
Ethiopia, Guinea-Bissau, Madagascar,
Rwanda, Cape Verde, Côte d'Ivoire,
São Tome/Príncipe, Seychelles, South Africa,
Then Mauritania, Mali, Namibia,
Congo of Brazzaville, Comoros, Cameroon,
That's all, there aren't any more!
This contains all 206 sovereign states as of September 2017, according to Wikipedia, and including even the disputed states. I did cheat a little in naming some of them. Don't @ me.
In theory, it preserves the rhythm and rhyme structure of the original song. Learning how to pronounce the names of every country in the world, and especially figuring out how I intended them to be stressed, is left as an exercise to the reader.
It's about 50% longer than the original, which named several things that aren't countries (Puerto Rico, Guam, etc.) and missed a whole bunch of small or very recently-created ones.
The original country analysis was done by some very dedicated people on the Animaniacs wiki.