gistにユーザ認証が成功したかどうか。fetchで。

まず、最も正しそうな方法。

https://api.github.com/authorizations
でチェック。ただし、basic認証のみ。

簡単な方法。HEADでチェック。

認証が成功するとapiの使用限界が変わる。60=>5000
あるいは status 401が認証に失敗したcodeとしてとりあえず返ってくる。
アクセス先は、gist内ならどこでもいい。
https://api.github.com/gists/public

 let chk=res.headers.get('X-RateLimit-Limit');
 //or res.status ==401 is NOT

全部

<input type="text"></input><input type="password"></input>
<button onclick="auth()">chk</button><label>ret</label>

let fn={}
fn.q=(d=>document.querySelector(d))
;
function isAuthGist(str){
  let url ='https://api.github.com/gists/public'
  ,opt ={method:"HEAD",mode:"cors",headers:{"Authorization":str}}
  ;
  return fetch(url,opt).then(res=>{
    let chk=res.headers.get('X-RateLimit-Limit');
    //or res.status ==401 is NOT
    console.log(res.status)
    return (chk && chk!=60)?true :false
  })
}
;
function auth(){
  let u=fn.q('input[type="text"]').value
  ,p=fn.q('input[type="password"]').value
  ,str ='basic '+ btoa(u+':'+p)
  ;
  isAuthGist(str)
    .then(d=>{fn.q('label').textContent=''+d;return d})
  ;
}
;

response header

/*NOT*/
Access-Control-Allow-Origin:*
Access-Control-Expose-Headers:ETag, Link, Retry-After, X-GitHub-OTP, X-RateLimit-Limit, X-RateLimit-Remaining, X-RateLimit-Reset, X-OAuth-Scopes, X-Accepted-OAuth-Scopes, X-Poll-Interval
Content-Length:93
Content-Security-Policy:default-src 'none'
Content-Type:application/json; charset=utf-8
Date:Wed, 07 Mar 2018 13:12:26 GMT
Server:GitHub.com
Status:401 Unauthorized
Strict-Transport-Security:max-age=31536000; includeSubdomains; preload
X-Content-Type-Options:nosniff
X-Frame-Options:deny
X-GitHub-Media-Type:github.v3; format=json
X-GitHub-Request-Id:F464:41E2:DE8869:104A905:5A9FE52F
X-RateLimit-Limit:60
X-RateLimit-Remaining:58
X-RateLimit-Reset:1520431935
X-Runtime-rack:0.061018
X-XSS-Protection:1; mode=block

/*authed*/
Access-Control-Allow-Origin:*
Access-Control-Expose-Headers:ETag, Link, Retry-After, X-GitHub-OTP, X-RateLimit-Limit, X-RateLimit-Remaining, X-RateLimit-Reset, X-OAuth-Scopes, X-Accepted-OAuth-Scopes, X-Poll-Interval
Cache-Control:private, max-age=60, s-maxage=60
Content-Encoding:gzip
Content-Security-Policy:default-src 'none'
Content-Type:application/json; charset=utf-8
Date:Wed, 07 Mar 2018 13:13:20 GMT
ETag:W/"ed8a0671388a83cfa6e5f705c5f2c191"
Link:; rel="next", ; rel="last"
Server:GitHub.com
Status:200 OK
Strict-Transport-Security:max-age=31536000; includeSubdomains; preload
Vary:Accept-Encoding
Vary:Accept, Authorization, Cookie, X-GitHub-OTP
X-Content-Type-Options:nosniff
X-Frame-Options:deny
X-GitHub-Media-Type:github.v3; format=json
X-GitHub-Request-Id:F464:41E2:DE94E1:104ACA4:5A9FE53A
X-RateLimit-Limit:5000
X-RateLimit-Remaining:4997
X-RateLimit-Reset:1520429607
X-Runtime-rack:0.237184
X-XSS-Protection:1; mode=block