Most lawfully present immigrants can buy health insurance through the new health insurance marketplaces. This group includes lawfully present immigrants who cannot enroll in Medicaid.
Note that in October 2018, the Trump Administration proposed an expansion to the current policy that determines whether certain immigrants would be considered a “public charge.” While the rule is not finalized, it may eventually impact individuals seeking admission to the U.S. or seeking to adjust status to lawful permanent residence (green card) applying for or using certain public benefits, like Medicaid. Under current and proposed rules, the application and use of premium tax credits and cost-sharing reductions are exempted from the public charge determination test. If the rule is finalized, you will have time to act before the rule goes into effect, and thus can still apply for and use public benefits including Medicaid in the interim.
Undocumented immigrants may not purchase coverage through the new health insurance marketplaces. (45 C.F.R. § 155.305).