Most people consider Greece as part of the West. Democracy evolved in many places across the West not just Greece. The Roman Republic is a more model closer to the representative democracy we have today than Athenian direct democracy.
If I meant the US, I would have said the US and not the West. The West includes most of Europe, the US, Canada, Australia and New Zealand.
Using "the West" to mean the US would be deeply misleading.
The US is both a Republic and a Democracy. This is an extremely common misconception among many Americans that a Republic is something different from Democracy. A Republic means you don't have a king. It has nothing to do with how leaders are elected in modern vocabulary.
Democracy as it exists today did not exist when the American founding fathers debated political systems. They called their system Republic as that is what the earlier Roman system was called.
However as European monarchies evolved into representative democracies, the concept and understanding of what a Republic and Democracy is has changed.
Both Norway and Finland are democracies for instance. What separates Norway and Finland is that in Norway we are a monarchy why Finland is a Republic. Instead of a King, the head of state is Finland is a President.
However in neither country does the head of state form goverenment. Memberes of parliament are elected and they form government.