Try using SimpleHTTPServer in Python.
mkdir ~/public_html
command_to_generate_output > ~/public_html/output.txt
(cd ~/public_html; python -c 'import SimpleHTTPServer,BaseHTTPServer; BaseHTTPServer.HTTPServer(("", 8080), SimpleHTTPServer.SimpleHTTPRequestHandler).serve_forever()')
The first two lines are setup for the web server. The last line creates a simple web server, opened on port 8080, which only serves filsefiles from ~/public_html. If only one file is in that directory, then only that is exposed: "http://localhost:8080/output.txt" http://localhost:8080/output.txt.