You can purchase residential or commercial properties in India 

2) NRI mutual fund investments are governed by the Foreign Exchange Management Act 1999 (FEMA). As per the provisions of the act, the NRIs are allowed to make investments in the capital markets including direct stocks, exchange-traded funds (ETFs) and mutual funds. These investments are allowed only with the compliance of certain conditions like fresh filing of Know Your Customer (KYC) documents, the opening of a rupee-denominated Non-Resident External (NRE) or Non-Resident Ordinary (NRO) account, etc.

There's no restrictions for a  NRI to purchase immovable property or invest in FDs or bonds in India other than purchasing agriculture property. 
The overall investment limit for NRIs/PIOs in stock investments is 10% of an Indian company's paid-up capital. Individual investment is capped at 5% of the paid-up capital.
Non-resident Indians (NRIs) do not need prior permission to buy or sell immovable property in India.
In India, NRIs are permitted to purchase both residential and commercial properties. However, NRIs are normally not allowed to own farmland, plantation property, or farmhouses unless they have a special exemption from the Reserve Bank of India (RBI) or meet specific requirements.

Yes they can invest 

For that you will have to check the visa terms whether or not there is any restriction 

However in my view there are restrictions placed by the RBI and Fema when money goes out of the country and NOT when money is being invested in India 

Voluntary Retirement' Can Be Sought Only If Officer Meets Eligibility Criteria; Resignation Can Be Anytime. 

H1B visa holders are Non-Resident Indians (NRIs) working abroad. They can very well purchase properties in India, and also invest in fixed deposits and other financial assets. RBI has allowed such investments under the FEMA.

Dear Client,

H1-B visa holders can purchase property in India without any specific restrictions. The acquisition of property in India is governed by the regulations set by the Reserve Bank of India (RBI) under the Foreign Exchange Management Act (FEMA). H1-B visa holders can invest in various financial instruments in India such as Fixed Deposits (FD), Public Provident Fund (PPF), Mutual Funds, etc. However, certain investment options like PPF have restrictions based on the individual's residential status. These investments are governed by regulations set by the RBI and other regulatory bodies such as the Securities and Exchange Board of India (SEBI) for mutual funds. 

- Being an NRI , you are only allowed to purchase residential or commercial property in India.

- However, as per FEMA21/2000 RBI rules , an NRI or PIO cannot acquire agricultural land /farm house/plantation property in India.

- If you will purchase an agriculture land in India then it will be a violation of FEMA laws , that can result in a penalty or even confutation of that property.

- Further, to purchase an agricultural property in India, an NRI has to take a prior approval from Reserve Bank of India.

- Further, as you are not attained the citizenship of US , then you can invest in banking instruments after opening an NRE account and also can invest in the stock market as well.