Basically what Murch said. You have to use the private key to sign the transaction.
There is a point I want to add, however. It is possible to send coins without importing the private key into a computer which is connected to the Internet. In fact, this is how a properly run exchange or online wallet service ought to maintain its cold storage. Normally, funds are just sent to the public address of the cold storage. When it comes time to send coins out of cold storage, the private key is imported into an offline computer, the transaction is generated and signed, and the signed transaction is then broadcast using a different computer (which is connected to the network). This increases security as it prevents malicious attackers from gaining access to the private key because they simply have no way of connecting to the offline computer on which it is stored.
Don't think a clean VM protects you against all attacks. The host (hypervisor) needs to be clean too, as it can access anything inside the VM too and do keylogging etc. Malware may not be sophisticated enough YET but I'm sure it will be sooner than later. – Jannes – 2014-03-21T22:19:15.453