I'm not aware of any specific US response to the Dubai threat... By the way, this was not made explicit in the "leg" statement, but rather in a separate IRGC statement, which also threatened Israel:
Iranian President Hassan Rouhani told the US that Washington might have "cut off the arm" of Soleimani but America's "leg" in the region would be cut off in response, Iran's Fars news agency reported.
[...]
The Islamic Revolutionary Guard Corps (IRGC) warned on their Telegram channel that in the event that Iranian soil is bombed, they would launch a third wave of attacks to destroy Dubai and Haifa.
Trump however has already warned Iran in general terms that the US will respond perhaps in a "disproportionate manner" to any Iranian actions, after striking Suleimani.
See also a related question in which the answer happens to touch on US public support for a US military response to the [alleged] Iranian attack against the Saudi oil facilities. (Yes, Trump might not be entirely bound by public opinion, but it's probably informative to what level of response he might choose in case non-US but allied facilities get struck in the Gulf.)
After the Iranian missile strike of Wednesday, which resulted in no casualties, Trump said the Iran "appears to be standing down". In that rather lengthy speech, he did not make any comments on IRGC's threats against Dubai or Haifa, as far as I can tell. He did also say
We are independent, and we do not need Middle East oil.
