See yourself. That's what Mistral answered:

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Question:

Where do the words "urbi" and "orbi" come from? Is there a connection to "urban" and "orbit"?

Answer:

The words "urbi" and "orbi" come from Latin and have different origins and meanings compared to the English words "urban" and "orbit."

Urbi: This word is the dative form of the Latin word "urbs," which means "city." In the context of "urbi et orbi," it specifically refers to the city of Rome.

Orbi: This word is the dative form of the Latin word "orbis," which means "circle" or "world." In the context of "urbi et orbi," it refers to the entire world.

The English words "urban" and "orbit" have different etymologies:

Urban: This word also comes from the Latin "urbs," but it has evolved in English to refer to urban or city conditions.

Orbit: This word comes from the Latin "orbita," which means "path" or "circle." It refers to the path that a celestial body takes around another celestial body, such as the Earth around the Sun.

In summary, there is no direct etymological connection between "urbi" and "urban" or between "orbi" and "orbit," although they all originate from Latin. The meanings and contexts in which these words are used are different.