The underlying problem is that your host has disabled curl, but not all of it, so WordPress detects that it is still enabled and attempts to use it. You can choose a different transport with a little minor code hacking.
First download this plugin and install it:
http://trac.wordpress.org/attachment/ticket/8086/list_transports.php?format=raw
After you activate it, you'll see a list of what transports WordPress detected as working at the bottom of the admin screens (in the footer). Once you know these, you can deactivate and delete the plugin.
Now, curl will probably be on the list, but you know that one doesn't work already. So pick one of the others and add this code to your theme's functions.php file:
function block_transport() { return false; }
add_filter('use_http_extension_transport', 'block_transport');
add_filter('use_curl_transport', 'block_transport');
add_filter('use_streams_transport', 'block_transport');
add_filter('use_fopen_transport', 'block_transport');
add_filter('use_fsockopen_transport', 'block_transport');
That will block all five possible transports from working. Now just remove the add_filter line on the transport that you want WordPress to actually use.