Even though there was no direct api request for getting uptime from newrelic we can give the nrql queries inside curl.
curl -X POST -H "Accept: application/xml" -H "X-Query-Key: your_query_api_key" -d "nrql=SELECT+percentage(count(*)%2c+WHERE+result+%3d+'SUCCESS')+FROM+SyntheticCheck+SINCE+1+week+ago+WHERE+monitorName+%3d+'your+monitor+name'+FACET+dateOf(timestamp)" "https://insights-api.newrelic.com/v1/accounts/your_account_id/query"
The above curl will give uptime percentages distributed by Days of week. If you don't know how to url encode your nrql query please try this http://string-functions.com/urlencode.aspx. Hope this helps.