A crude, viable explosive device was discovered during a security operation in the Coolnagard area of the Co Tyrone town on Saturday.
It followed an incident the night before when the officer living at the house fired his gun.
The Police Federation for Northern Ireland said the device was "hurled into his home" but failed to explode, and branded those responsible as "cowards".
No-one was hurt and the incident has been referred to the Police Ombudsman.
An investigation has been launched into the discovery of the bomb and officers have asked anyone who witnessed suspicious activity in the area in recent months to come forward.
Police have been going door-to-door in their efforts to glean information.
"People, especially in the Coolnagard area of Omagh, who saw anything that gave them cause for concern or they thought out of the ordinary over the past couple of months and specifically between 8pm and midnight on 25 January to contact police," a statement said.
"Police are also asking if anyone saw anything that raised their concerns in the Kevlin Road area of Omagh during these times, again to contact Police. Police would like to thank local people for their patience and co-operation during this security alert."
Information can be passed on by calling 0845 600 8000.
A number of homes were evacuated as searches linked to the bomb took place.
West Tyrone Ulster Unionist MLA Ross Hussey, who sits on the Northern Ireland Policing Board, said the attack was a sinister reminder of Northern Ireland's troubled past.
Mr Hussey said: "Omagh and Northern Ireland don't need this and we won't let morons drag us back to the past.
"Coolnagard is a quiet residential area where everyone gets on very well. Those who carry out such sinister acts are only trying to drive a wedge between communities. Now is a time when we should all stand shoulder to shoulder against terrorism, from wherever it may arise.
"Once again, we should also acknowledge the work of the security forces including the PSNI and Army Bomb Disposal team who responded promptly in the aftermath to seal off the area, evacuate residents and deal with the explosive device."
Sinn Fein MLA Declan McAleer said he was deeply concerned.
He said: "I spoke to several of the evacuated residents and they are very concerned that such a sinister incident involving the discharge of a firearm and a suspicious device has taken place in their community. I am therefore relieved that no one has been hurt or injured in this incident.
"Coolnagard is a very quiet residential area of the town where people from all sections of the community live side by side and where community relations are very good.
"The investigation is still ongoing so I await further details from the PSNI as to the exact nature of the events which unfolded here on Friday night."