So I invested some time and this what I've found
1/ You should have your custom search engine
https://cse.google.com
2/ Configure it to search all webs
https://support.google.com/customsearch/answer/2631040?hl=en
3/ Make query to that engine following the document
https://developers.google.com/custom-search/json-api/v1/using_rest
Spelling / Did you mean repairs if possible can be found in "spelling" section. (Irrelevant sections removed) . Fox ex: I searched for "devart" here
{
"kind": "customsearch#search",
"url": {
},
"queries": {
"request": [
]
},
"context": {
},
"searchInformation": {
},
"spelling": {
"correctedQuery": "deviantart",
"htmlCorrectedQuery": "<b><i>deviantart</i></b>"
},
"items": [
]
}
I had the same question and its downvoted also while they are allowing these questions on the site. You may want to read about the auto-complete apis here
https://stackoverflow.com/questions/40061449/google-auto-correct-api
Google Search autocomplete API?