Unfortunately I have had experience of a real bullying case. Legal remedies are virtually impossible under Irish law. There is unfair/ constructive dismissal legislation, but bullying per se is not illegal. A senior barrister told me that 'employers will rather admit to rape than to bullying' so don't assume that alleging bullying is an easy way for the employee to make a quick buck.
It's not as black and white as that. The Safety, Health and Welfare at Work Act, 2005 has a duty on the employer to prevent "inapropriate behaviour" and employees must not engage in "inappropriate behaviour". Given the accepted definition of bullying includes reference to "inappropriate behaviour" there is a clear legislative link to prevent and not be involved in bullying. The HSA's most recent Code on Bullying would further qualify this link.
However, the key aspect is a fair investigation following any complaint, especially one that is formalised in the form of an official letter of complaint. The investigation must be based on one of natural justice as any internal investigation. The accused individual is rightly assumed to be innocent of the complaint until the accuser can satisfy, with evidence, that their complaint is valid on the balance of probabilities.
To that extent if there is no investigation, even though a formal complaint was made, this would be unfair the the individual accused. It is a serious accusation and for the OP may have further consequences in the future if the claim is left on file and they haven't had the opportunity for it to be dismissed.
The HR unit should have a policy and procedure in place regarding the investigation and handling of any bullying complaint. I'd start there and insist that the complaint is investigated (though some policies do include a suspension during this period) if it hasn't, plus also look at the appeals procedure that shoudl also feature in the policy.
I'd agree go into it calmly, but under the context of current standards, it is a serious allegation and the OP deserves to have the complaint looked into because it is potentially a serious slur on their character and capabilities.