I have an API URL of SMS Gateway :
How can I make REST API using this gateway, so can call the API in my mobile app?
You need to call this rest API in code on each event of your requirement like New order, order update, Here are the ways you can use this API:
Solution 1 :
Go into code, Place API in code of each event Ex. Checkout, Order Update which require code knowledge in deep.
There are free sms notification modules available for magento like "Spring Edge Magento sms extension" which is Compatible with: Magento versions 1.x and 2.x
Magento sms module events configuration :