A police patrolman pulls a car off to the side of the road, perhaps for a defective tail light. The driver arouses suspicion for any number of reasons -- perhaps spent pistol cartridges on the floor -- and the policeman asks the driver to open the trunk of the car. The driver refuses. If there is sufficient reason to do so, the patrolman can get a court order and have the trunk opened. The motorist has no right to refuse the court order.
Police don't need a warrant to search your car. They only need probable cause.
If they don't have probable cause, they can also ask you to allow the search. You don't have to agree, but you have the option.
Replace the patrolman, the motorist and the court order with the House of Representatives, the president and a subpoena. There's a certain degree of similarity, nu?
No.
Impeachment is not a criminal process. It's a political process to determine if an official has engaged in conduct that justifies removal from office.
The Constitution is very open-ended about impeachment. It does not stipulate any sort of due process rights, certainly not during the investigative phase. While it is a good idea to give the impeached official the ability to respond to charges during the Senate trial, there is no requirement to do so under the Constitution. The House could impeach on Monday, and the Senate could convict on Tuesday, if they chose to do so.
The federal government is not a private citizen, and is not protected by the 4th Amendment. Everything the federal government does, with the exception of anything classified or specifically shielded by executive privilege, is supposed to be tracked, recorded, archived and available to the public.
If there is an individual working for the federal government who believes that their statements to Congress may be incriminating, then they can invoke their 5th Amendment rights. However, federal agencies don't have 5th Amendment rights, only individual staffers.
Congress has the power to issue subpoenas for its investigations. The only real objection the White House has right now is to claim executive privilege, which is not a blanket excuse to avoid accountability. Some things will reasonably be shielded, but most of it won't.
The Executive Branch has the option to fight Congress' requests in court. So far, they haven't convinced many courts that their arguments are sound.
The Executive Branch has also ordered multiple officials not to comply. Several officials have declined, and so far Congress has not chosen to take their refusal into court -- instead, it's likely going to be treated as evidence of obstruction. Despite that demand, numerous individuals have voluntarily appeared.
In other words: Congress is acting 100% in accordance with the law.
