Can the European Union refuse a proposed ambassador
Of course it can.
The sending State must make certain that the agrément [sic] of the receiving
State has been given for the person it proposes to accredit as head of the
mission to that State.
Vienna Convention on Diplomatic Relations (1961). p.100. Retrieved from U.N.
So if, for example, some other country proposed as ambassador to the USA someone who declared their aim as being the destruction of the USA, The US government could veto this appointment.
In rare and perhaps more serious circumstances an existing ambassador is removed:
Article 9. The host nation at any time and for any reason can declare a particular member of the diplomatic staff to be persona non grata. The sending state must recall this person within a reasonable period of time, or otherwise this person may lose their diplomatic immunity.
Vienna Convention on Diplomatic Relations (1961). Retrieved from Wikipedia
On 5 August 2018, the Canadian Ambassador to Saudi Arabia, Dennis Horak, was declared persona non grata after Canada called for the release of civil activists in detention.
List of people declared Persona Non Grata, Retrieved from Wikipedia
EU and Vienna Convention
The EU is not a nation state, but acts like one in this regard.
The European Union has a unique sui generis status on the international plane, which is reflected in its capability to enter into diplomatic relations with third states and international organizations. Over nearly six decades, the European Union (EU) has gradually built its own worldwide bilateral and multilateral diplomatic network, which is made subject — through specific agreements with the host country — to the 1961 Vienna Convention on Diplomatic Relations.
The EU and International Diplomatic Law: New Horizons? (2012). Hague Journal of Diplomacy. Retrieved from Brill
I imagine it would ultimately be the head of the executive branch† of the European Union. The president of the European Commission.
REQUEST FOR APPROVAL OF THE APPOINTMENT OF A HEAD OF MISSION
The procedure, which implies the approval from the European Commission, the Council of the European Union, the European External Action Service and the authorities of the Member States of the European Union, generally lasts eight weeks.
[Handbook] for the use of the Diplomatic Corps accredited to the European Union. Retrieved from Europa.eu
Similar to a state, the EU has a legislative branch (Parliament + Council), executive branch (Commission) and independent judiciary (Court of Justice).
Institutional Affairs. Retrieved from Europa.eu