Combining @Byron_JMP's Web Browser Box idea with your regex string you could save a formatted html file with embedded links and display that in your window.  Suddenly you have a lot of control over the text formatting; you could even let users optionally add a friendly name to links syntax like [JMP Website]http://www.jmp.com, similar to some markup syntax.
 

 
Names Default To Here( 1 );
text = "Periarthritis is defined as: Inflammation of the tissues around a joint. " ||
"[A Link]http://www.reference.md/files/D010/mD010489.html  As arthralgia, myalgia, tendonitis " ||
"and other musculoskeletal issues are listed events, periarthritis can be considered listed as well. 
Comment: no good sentinel case — not a new signal; one case also reported somnolence and " ||
"another case reported “yawny” which could mean sleepy *http://www.ncbi.nlm.nih.gov/pubmed/12530994";
t = text;
//wrap links in a string with <a href> tags
findlinks = function({t, recct=0}, {url_regex, rgxmatch = {}, rtn=""}, 
	recct++; 
	if(recct > 100, 
		throw("Recursion to deep: " || char(recct) || " levels.  Still need to parse: '" || t || "'"));
	url_regex = "^(.*?)(\!\[(.+)\])?((https?://){0,}(\w+\.){2,}\w+/?(\w+/?){0,}(\.\w+){0,1}/?)(\b.*)$";
	
	rgxmatch = Regex Match(t, url_regex);
	if( N Items(rgxmatch) == 0, 
		rtn = t, //No link
		if( N Items(rgxmatch) == 10,  //found a link 
			rtn = rgxmatch[2] || 
				"<a href='" || rgxmatch[5] || "'>" || if(is missing(rgxmatch[4]), rgxmatch[5], rgxmatch[4] ) || "</a>" ||
				findlinks(rgxmatch[10], recct),
			throw("Error matching regex in: '" || t || "'")
		)
	);
	return(rtn)
);
//Create a simple html page from a block of text, save it to the temp files, and return the filename
texttohtmlfile = function({t},
	Save text file("$TEMP/Temp HTML file.html", "<html><body>" || findlinks(t) || "</body></html>")
);
//Make a temp html file
fn = texttohtmlfile(text);
//Simple window to display html file, deletes the temp file when it closes
w = New Window("Example",wb = Web Browser Box());
w << On Close( Delete File( fn ) );
wb << Set Size(1000, 200);
wb << Set Auto Stretching( 1, 1 );
wb << Set Max Size( 1000, 1000 );
//Show the html file in the window
wb << Navigate( fn );
Note the unfortunate slash bang in the regex match expression as "\[" begins an 'escaped character' section.
 
Edited to add screenshot and again to fix a typo.