It is illegal in all states to knowingly file a false police report.
However, your husband would have no legal recourse against the filer of the false report unless he was investigated by the police, arrested after being investigated, and then charged or prosecuted based on the false allegations made in the report. If cleared of all charges, he could then go after the person who filed the false report.
The likelihood of your husband being arrested and charged with a crime based on a false report, after an investigation was conducted, would be slim - as the police would have no evidence to support such an arrest.
It is up to the state to take any action against the filer of the false report otherwise. The state would need to show that the person filing the report filed it knowing it was false, which is not an easy proof to make. Unless the police investigation was time consuming and costly and the results showed clearly that the person reporting the crime did so knowing it was false, there is little likelihood that the state would pursue any action.
Phone companies keep all phone records, although there are those who use pretexting to gain access to records and phone records can be obtained illegally for a price. Generally customer records will not be released without a court order, however.
Phone logs can be deleted, which alters the log, but logs cannot be edited and individual call entries cannot be modified. The phone records record call numbers, dates, times and durations.
Call spoofing or caller ID spoofing can cause a phone to display a number that has been programmed in, which could allow for your husband's number to appear in the recipient's call logs. Right now, neither call spoofing nor caller ID spoofing is against the law, although the Massachusett's legislature is considering making it illegal, and there was a bill introduced in Congress this January that patterns itself after a "Truth in Caller ID Act" that was originally introduced in 2007 and died in committee.
In 2008, a Pennsylvania man was arrested for making threatening calls, and it was discovered his number appeared on the recipient's caller ID as a result of caller ID spoofing.
If you think your husband could be a victim of call spoofing, he could call the FCC at 1-888-255-5322 or he could call your state Attorney General's office.
Edit to add:
I am not sure which of the three posts you started you will be looking at, Renee, but I thought I would include another phone number here that your husband could call: FTC at 1-877-382-4357. In addition, I mentioned this on another one of your threads, but it is entirely possible that the person who filed the report on your husband was a victim of spoofing as well, and legitimately thought the threatening call originated with your husband. If this is the case, the police report would not be considered a "false" report as it was made in the belief that what was reported was true.