“A denominational hospital must not direct or otherwise cause a registered health practitioner … to refuse to provide mandatory advice and services on the basis of a conscientious objection,” it reads.
The push follows the US Supreme Court in June reversing its landmark Roe v Wade decision to guarantee abortion as a constitutional right.
Prime Minister Anthony Albanese was asked on 20 July if he would force publicly funded hospitals to offer abortions and he shot down the suggestion.
“These matters are between a woman and her doctor,” Mr Andrews said.
If it is passed unamended, the changes would come into effect from July next year.