The fundamental principle of Democracy is rule by people whereas that of Islam is rule by Allah (swt) alone. Allah (swt) is the only one who has the ultimate authority to legislate and in Democracy this authority is given to people.

There are countries where Muslims are a majority, however not necessarily Islamic as Western laws are implemented as opposed to Shariah as the law of the land. Being Democratic, the elected representatives, although Muslim, may pass a law making Halal what Allah (swt) made Haram and vice-versa. Meaning introducing man made primary laws as explained by Dr. Bilal Philips in this module.

For example (this was mentioned by Dr. Bilal Philips in another class): The Prime Minister of a Muslim country a few years ago changed the law of inheritance from being women receiving half shares than that of men to equal shares for both. As this law was passed through parliament by means of voting by the elected Muslim representative, it is the essence of Democracy; however this is clearly not Islamic. Wouldn't this be considered an act of shirk / kufr?

We need to distinguish between an act of Kufr and a person being declared a disbeliever. In general, yes, to deny or change the inheritance laws of Islam is Kufr.

Sadly, many ignorant Muslims today (not just governments) also deny these laws and demand unislamic shares of the inheritance. We do not make Takfeer of them but we warn them that such a belief and act is Kufr.

Allah knows best