A tire with the size of 700 x 23c, when changed to inches, what is the diameter and what is the width?
You're asking a question that cannot be answered. Sorry - the plethora of tyre size measuring systems means there is no good answer.
Diameter could be 29 inches, it could be 28 inches, it could even be 27 inches. But the bead seat IS 622 millimetres without exception.
The width likewise varies - there are fractional inch measurements that do not line up with the decimal inch of the same size - IE 1 1/2 inches is not the same width of tyre as 1.5 inches. But 23mm is 23mm without exception.
This is why the ETRTO measurement system is simply superior to all that went before. Not that its based on metric, but ETRTO is consistent and accurate.
A "700c" tyre in a width of 23 is a ETRTO measurement of 622-23, which is sometimes written as 23-622.
You might glean some more information from How do I know what size tyres can I fit on my rims? and from How do I determine my wheel size
Or consider asking about what you're trying to achieve - this sounds like an XY problem.
To answer the question as written, 23 is the width of the tire in millimeters when mounted and inflated. The actual width depends on the rim width, and manufacturers typically don't tell which rim width the announced width is measured with.
Diameter is more complicated. 700mm is the outer diameter of "C" type tire when mounted and inflated. This gives rim diameter of 622mm (the same as 622, one of the two 28"s and 29"). It does not make any sense, but you can memorize or look up it. The 700A and 700B sizes have been long forgotten, but 650A and 650B are specified the same way.
It is correct (as stated in the other answers, which I've voted up) that tyre sizing systems mean the actual stated sizes can't be converted, and in fact the width of a given tyre will vary depending on the rim width.
That said, ETRTO 622-23 tyres are sometimes sold as 28x0.9" (random web example). This link is priced in €, which isn't surprising - it's often German manufacturers/sellers that use 28" as a tyre size, even though they don't use inches otherwise. This may help you if you're shopping for tyres and they're only specified in inches, but confirm the ETRTO size.