New immigrants to USA are NOT eligible for benefits like Medicare.

Medicare has two parts: Part A and Part B.
Part A covers major things like hospitalizations, surgeries etc.
Part B covers other things like prescription drugs, doctor office visits etc. Within Part B, there are different levels of coverage: From A to J.

Regular US persons (citizens, residents etc.) can get Medicare Part A if they have worked in US for 40 quarters (10 years for most people) and are above the age of 65+.
New immigrants can't get that. If they have stayed in US for 5 years as a green card holder, and above 65+, they can BUY medicare from the government.

Part B has to be purchased by everyone, even those who are US citizens, born in US and have worked in US all their life. This is commonly referred as Medicare Supplement Insurance. (Again, the eligibility to buy this one is also same as Part A.)

In short, new immigrants can't get Medicare. Therefore, they will have to purchase private health insurance. However, as everyone in US who has been here for long time will go to Medicare at the age of 65, regular domestic health insurance companies like Aetna, Cigna, Blue Cross, Fortis, Unicare (and thousands others) don't sell health insurance to 65+. For those below 65, depending upon your state, and depending upon how long you have been living in US, you might be eligible for some domestic health insurance plans.

New immigrants are entering the system from the middle. Then how to fill the gap?

Therefore, new immigrants have to purchase essentially the same type of insurance plans that are purchased by visitors. Actually, visitors insurance is a popularly used term. However, it is the same temporary insurance for anyone who comes to the US, new or temporarily on any visa, or even new immigrant, who is a non US citizen, that can't get insurance from regular domestic health insurance companies in US.
Only difference is the duration for which people would want to buy the coverage. While visitors would buy coverage for a short period, new immigrant would buy for a longer duration. Keeping that in mind, we have dedicated a page just for new immigrants health insurance.

Of course, any such gap filling insurance would NOT cover any pre-existing conditions, general physical check ups etc. It is like this, "something is better than nothing". You are buying it because there is no other option.

Also, you can't get other benefits like Medicaid etc. Before they immigrate to the US, you would have signed an affidavit of support form I-864 that you are going to take care of all their expenses and they will not be a burden on the government etc, etc. I864 is a legally binding contract that can be enforced and should be taken seriously.

If they now try to utilize anything like Medicaid because of old age, no or low income etc, government can actually send the bills to their sponsor for reimbursement. This voids the purpose of trying to utilize anything like Medicaid and it may also adversely affect your ability to sponsor more people in future.