Did the US give permission/support for Iraq to invade Kuwait?

No, that's not accurate. The Bush administration certainly didn't give Saddam the go ahead to invade Kuwait, although that doesn't mean that Saddam didn't interpret that the US had given him the go ahead. The British ambassador in the documentary you're referring to is probably referencing the infamous "Glaspie meeting," named after the US ambassador to Iraq, April Glaspie. On July 25, 1990, Glaspie met with Saddam to discuss recent Iraqi military mobilization along the Kuwaiti border. While the several different records of this meeting are contradictory, we do know that Glaspie told Saddam something along the lines of "We have no opinion on Arab-Arab conflicts." This, some have argued, represented a tacit American approval for Iraq to invade Kuwait.

What this conclusion ultimately leaves out are other US diplomatic actions between the initial Glaspie meeting and the Iraqi invasion of Kuwait, and these subsequent actions show a much more complicated picture (as a sidenote, I've done some research on this topic at the George H.W. Bush presidential library, so I will be using primary rather than the secondary sources for this post). For instance, five days after the first Glaspie meeting, on July 30, Glaspie again met with Saddam. Relaying the president's and secretary of state's words, Glaspie told Saddam:

The United States and Iraq both have a strong interest in preserving the peace and stability of the Middle East. For this reason, we believe that differences [between Iraq and Kuwait] are best resolved by peaceful means and not by threats involving military force or conflict.

Admittedly, this statement isn't exactly a clear warning that if Saddam invaded Kuwait, the US would respond with military force. It does, however, show that the US did not view military aggression as an acceptable way for Iraq to handle its dispute with Kuwait. The worst that can really be said about the Bush administration was that it failed to clearly tell Saddam that the US would respond with military force to an invasion of Kuwait.

The Bush administration also had no reason to give Saddam a go ahead because the president and most of his foreign policy advisers were well aware of the negative consequences such an invasion would have for US national security. The day after Iraq's invasion, the Bush administration's National Security Council convened to discuss how the US should respond. During the meeting, national security adviser Brent Scowcroft stated that the failure or the unwillingness of the US to push Iraq out of Kuwait would allow Iraq to "dominate OPEC policies…and lead the Arab world to the detriment of the United States, and the great stakes we have in the Middle East and Israel.” The expression of such concerns by top-level Bush administration officials, I think, shows that the US never gave a go ahead for Iraq to invade Kuwait. Why would the US support an action that just a few days later they viewed as a significant threat to American national security? It doesn't add up.

Far more likely, the Bush administration simply didn't believe that Iraq would actually invade Kuwait. Glaspie and many other top Bush administration officials have admitted this. Moreover, the US was attempting to mend relations with Iraq before the Gulf War in the hopes that Iraq would become a key regional ally (along with Israel) that could contain Iranian aggression. In fact, Iran was seen was a much bigger threat than Iraq before the Gulf War. These two reasons--the belief that Iraq wouldn't invade and the desire to improve relations with Iraq--are, in my opinion, much more plausible answers as to why the US did not clearly tell Saddam that the US would go to war over Kuwait. Consequently, because the US failed to clearly state its position on the matter, Saddam (mistakenly) assumed that the US wouldn''t care too much if Iraq invaded Kuwait.