The Springwatch presenter said he had contacted police over the incident, which also saw his gate damaged.
A Change.org petition titled "BBC to sack Chris Packham" has over 70,000 signatures.
Mr Packham said: "All it does is strengthen my resolve to make the United Kingdom countryside a better place for wildlife and the people who live and work there".
While the Government agency, Natural England, stated: "Hello @ChrisGPackham - of course we don't condone this type of behaviour, it's never justified no matter how strongly people feel about an issue".
Announcing the decision to revoke the general shooting licences, Natural England said it would look to bring in "alternative measures" over the coming weeks to allow the lawful shooting of the bird species to continue.
Landowners have argued that this has come at the worst time of year, as it is when birds are beginning to lay their eggs, and those taking care of the rare species have had no time to prepare or apply for new individual licenses.
Explaining the licencing changes, Natural England said that following the legal challenge on the way licences are issued, users who rely on them are not acting lawfully.
The ban has been criticised by members of the British Game Alliance and many farmers, who say that pigeons need to be controlled in order to protect crops.
Wild Justice launched its legal challenge on the basis that Natural England could not be satisfied that conditions required to legally kill 16 bird species without having special licences were being met. "Hundreds of jobs are at severe risk if a solution isn't found".
Under "general licence" rules certain birds are effectively treated as vermin and can be killed, while there is a general principle in law that all wild birds are protected under the Wildlife and Countryside Act.
A Hampshire Constabulary spokesman confirmed a report of criminal damage has been made with the incident believed to have taken place between 20:00 on Wednesday and 06:50 BST on Thursday.
Anyone with any information was urged to call police on 101 quoting 44190141581.