We don't tend to use "father" as a proper noun in this way. When Dad is used as a proper noun in place of someone's name it is always being used to one's own father. Using the formal term "Father" as a proper noun sounds stilted and stiff. It seems like inappropriate formality.
Learners will rarely need to use "Dad" as a proper noun, and will never need to use "Father" like this. It is possible to say "How is Father's health" (capitalised) but a context in which you should say this will rarely arise.
If you were speaking to the doctor you might say "How is my father's health"? If you were speaking to your sister you might say "How is Dad?", although an English Learner would not usually speak to their own sister in a foreign language. If you were writing an upper-class character from the 1920s, you might have that character say "How is Father?"