If the baby has heart disease: The girl whose baby has heart disease is American. Or, The girl, whose baby has heart disease, is American.

If she has it: The girl with a baby and heart disease is American.

Punctuation is the key to making the meaning clear.

However this is a better sentence.

The girl, whose baby has a heart disease, is American.

The girl, with a baby, who has a heart disease is American.

The girl, with a baby who has a heart disease, is American.

Notice I also capitalized “American”. It is a proper noun.

the girl with the baby, who has heart disease, is american. – the girl is american with heart disease

or

the girl with the baby who has heart disease, is american – the baby has heart disease, the girl is american.