Save bookmarklet as "target"/"location" for a new browser bookmark, then click on bookmark to send current HTML to W3C for validation (see below for non-minified code)

validate-current-html-bookmarklet.html

javascript:var%20doctype=document.doctype?(new%20XMLSerializer).serializeToString(document.doctype)+"\n":"",content=doctype+document.documentElement.outerHTML,iframe=document.createElement("iframe");iframe.style.position="fixed",iframe.style.zIndex=1e4,iframe.style.top=0,iframe.style.left="10px",iframe.style.width="calc(100%%20-%2020px)",iframe.style.height="calc(100%%20-%2020px)",iframe.style.border="3px%20solid%20red",iframe.style.background="white",document.body.appendChild(iframe);var%20target=iframe.contentDocument||iframe.contentWindow.document;target.write("loading..."),target.close();var%20xhttp=new%20XMLHttpRequest,fd=new%20FormData;fd.append("content",content),xhttp.onreadystatechange=function(){4==this.readyState&&200==this.status&&(target.write(this.responseText.replace('href="style.css"','href="https://validator.w3.org/nu/style.css"')),target.close())},xhttp.open("POST","https://validator.w3.org/nu/#textarea",!0),xhttp.send(fd);

validate-current-html.js

// get current HTML code including DOCTYPE
var doctype = document.doctype ? (new XMLSerializer().serializeToString(document.doctype) + "\n") : "";
var content = doctype + document.documentElement.outerHTML;

// create iframe for displaying the result
var iframe = document.createElement('iframe');
iframe.style.position='fixed';
iframe.style.zIndex=10000;
iframe.style.top=0;
iframe.style.left="10px";
iframe.style.width="calc(100% - 20px)";
iframe.style.height="calc(100% - 20px)";
iframe.style.border="3px solid red";
iframe.style.background="white";
document.body.appendChild(iframe);
var target = iframe.contentDocument || iframe.contentWindow.document;
target.write("loading...");
target.close(); 

// prepare send POST request to W3C validator
var xhttp = new XMLHttpRequest();
var fd = new FormData();
fd.append('content', content);
xhttp.onreadystatechange = function() {
  if (this.readyState == 4 && this.status == 200) {    
    // replace relative path to W3C stylesheet
    target.write(this.responseText.replace(
        'href="style.css"',
        'href="https://validator.w3.org/nu/style.css"'
    ));
    target.close(); 
  }
};
xhttp.open('POST', 'https://validator.w3.org/nu/#textarea', true);
xhttp.send(fd);