LAX has regular mobile phone service, as well as pay phones and Internet access.
In order to connect to a mobile phone network, anywhere in the world, your device must support the frequencies in use. While there is no overlap between U.S. frequency bands and Indian frequency bands, it is entirely possible that your device can support frequencies in both countries. For example, the Apple iPhone 6 model A1549 (sold internationally) can connect to most 2G GSM, 3G UMTS, and 4G LTE networks in both India and the U.S.
If your device does not support U.S. frequencies, you will need to use Internet-based communication, or find another alternative. This is a hardware limitation, not software, and so replacing the SIM card or downloading an app will not help you.
When you turn on your phone after arriving at LAX, it will try to connect to an available carrier.
Your phone should indicate if the connection is successful; you may also receive an SMS welcome message from the carrier. Either way, you should be able to make domestic calls and send SMS messages, prefixing the number with +1 as the "country code" for North America.
You will be "roaming" internationally, which can incur some heavy tolls depending on your plan, so check on those rates before you leave and check which carrier you are connected to before placing any calls or sending any texts. This is particularly important with data, as after many hours in flight a smartphone may demand gobs of data as it syncs if you have not disabled international roaming.
If you have a compatible phone and wish to make regular calls but do not wish to incur roaming charges, you can purchase a prepaid SIM card from a U.S. carrier. ICE Currency Exchange booths sell SIM cards; there may be other vendors as well. The prices are likely to be exorbitant, however. If you do not plan to use the service over the course of the trip, you might look into acquiring one before you leave, or using alternative communications.
Pay phones are available at all LAX terminals, but mostly airside. There are some in the ticketing area at the Tom Bradley International Terminal (TBIT) as well; consult the LAX Terminal Maps. If they're like the pay phones in my area, I would expect to pay 75¢ or $1 for a local call, payable in quarters (pay phones that accept credit cards exist, but are rare). There was also a long distance surcharge for calls placed to lines farther than about 12 miles away, but it has been many, many years since I've used a pay phone and I don't know how any of that works any more.
The public areas at LAX have free WiFi, but as with most airport networks, the pipes are often overloaded. Even if you are able to connect to the airport, the bandwidth and stability may not be adequate to make VOIP calls, even if VOIP and video services are not explicitly blocked/banned.
Lastly, for true emergencies, the people at the information desk may be willing to let you use their phone as a courtesy, likewise phones in any airport clubs/lounges you may have access to.