Is there a law or regulation that prevents the use of dogs to detect the coronavirus at airports in the United States as it is done already somewhere else? Do other countries have laws like these which block the use of dogs?
The issue is more likely that there is no law or regulation allowing the use of dogs to sniff passengers in the US for matters not related to crime, whereas Dubai, a monarchy, has a different political system. In general, you have the constitutional right to move about in the U and the right not to be unreasonably seized, a right which may not exist in some other jurisdictions. That right can be restricted in accordance with law, but there has to be some such law. It is not a crime to have covid, or to travel while infected (it is a crime to carry bombs and drugs on a plane).
One area where there is some restriction is where there always has been a restriction, namely entry into the US. There is (was) covid screening at certain airports for flights from certain countries. This directive indicates the airport restrictions, but does not explain what screening will take place. The above DHS site says that "the passenger will be asked about their medical history, current condition, and asked for contact information for local health authorities. Passengers will then be given written guidance about COVID-19 and directed to their final destination, and immediately home-quarantine in accordance with CDC best practices". It is not clear whether a mandatory temperature check without even reasonable suspicion of a crime would constitute an unreasonable search (body searches are subject to higher standards than property searches, it seems), and constitutional law surrounding searches has emphasized the primacy of privacy in search law, not the fact of physically entering property of the body. Some airlines and airports offer voluntary temperature checking, so it might be possible if someone has a covid-sniffing dog to offer voluntary sniff testing, especially if it is offered by a private concern and does not have the appearance of government mandate (which would require a law).
The other answer speculates "that there is [likely] no law or regulation allowing the use of dogs to sniff passengers in the US for matters not related to crime," but that is not the case. While the use of dogs for health screening does not appear to be mentioned explicitly in US law, there are certainly laws that could permit it. If the administration claims authority to employ dogs for this purpose under these laws, they will be able to do so unless and until someone challenges the administration's interpretation successfully in court.
Congress has authorized the surgeon general (at 42 USC 264(a)) "to make and enforce such regulations as in his judgment are necessary to prevent the introduction, transmission, or spread of communicable diseases from foreign countries into the States or possessions, or from one State or possession into any other State or possession."
Regulations under that authority include 42 CFR 71.36(a):
The Director [of the Centers for Disease Control] may require that an individual arriving into the United States undergo a medical examination as part of a Federal order for quarantine, isolation, or conditional release.
Now a "medical examination" must be conducted by an "authorized and licensed health worker," but there's nothing anywhere that prevents the health worker from ordering the patient to be screened by a dog.
Similarly, at 42 CFR 71.32(a):
Whenever the Director has reason to believe that any arriving person is infected with or has been exposed to any of the communicable diseases listed in an Executive Order, as provided under section 361(b) of the Public Health Service Act, he/she may isolate, quarantine, or place the person under surveillance and may order disinfection or disinfestation, fumigation, as he/she considers necessary to prevent the introduction, transmission or spread of the listed communicable diseases. Executive Order 13295, of April 4, 2003, as provided under section 361 of the Public Health Service Act (42 U.S.C. 264), and as amended by Executive Order 13375 of April 1, 2005, contains the current revised list of quarantinable communicable diseases, and may be obtained at http://www.cdc.gov and http://www.archives.gov/federal-register. If this Order is amended, HHS will enforce that amended order immediately and update this reference.
So, if the president were to issue an executive order identifying COVID-19 for the purpose of this section, the authority to "place the person under surveillance" would enable screening with dogs. (I suspect, however, that the president has been too busy focusing on immigration law, which explains why he is using immigration law instead of public health law for public health purposes. At any rate, I am unaware of any such order or any orders related to 42 CFR 71.40, prohibiting the introduction of persons from designated foreign countries and places into the United States.)