Can we use the indefinite article before the name of a day of the week?
I was born on a Sunday.
I was born on Sunday.
Which one is more suitable and correct?
You should use the indefinite article, otherwise you run the risk of suggesting you were born last Sunday.
Interestingly, when we talk about months, we do not use indefinite articles, despite the fact that months seem to be quite analogous to days (they are both names or indexicals used recurringly to denote intervals of time). One does not say, for example, "I was born in an August." Why this should be the case is an interesting question and may have to do with the difference in prepositions used for days ("on") versus months ("in").
The way I understand is that any indefinite article like 'a' or 'an' is used with the names of the days of the week when it is not referred to a specific day. Just like your example 'I was born on a Sunday'. Here in this example Sunday is not any particular Sunday; this Sunday could be any Sunday.
Now, if you use the second example (without article), that will probably mean you’re born on last Sunday at the time of your statement. So if you don’t want to mean that you’re born on last Sunday or something like that, then you should go with the first example. In the comment section WS2 too contributed the same idea.