"whether" and "either" are two different words pragmatically. There is a slight difference between them, and they are not quite the same.
For a start, both are fixed expressions, which means you should be careful not confuse them with one another.
- whether ... or ... ( meaning: it is not important if );
whether ... or not ( meaning: if, or not );
- either ... or ... ( meaning: there is a choice between two possibilities );
Now that I've covered the differences, you can decide which to use and when.
He seemed undecided whether to go or stay. (a doubt between alternatives);
It's fine to do either.