Upvote:3
Australia offers ETA for citizens of several countries, including the US, but not including India. Having permanent residence in one of those countries does not suffice; you must have a passport issued by a qualifying country.
Australia also offers eVisitor for citizens of several countries, not including India, so that also won't help. You therefore need a visa for Australia.
New Zealand offers a visa waiver for citizens of several countries, including the United States, but not including India. Eligibility depends on being a citizen of one of the visa waiver countries. US non-citizen nationals are included, but not permanent residents. You therefore need a visa for New Zealand.
You can qualify for the programs mentioned above if you naturalize in the US and travel with a US passport.