Can European Union refuse a USA Ambassador proposal? If yes, who decides?

Question

This article tells us about the likely EU ambassador, Ted Malloch's not so politically correct declarations about the European Union:

Donald Trump's likely pick for ambassador to the European Union has suggested he wants to bring down the bloc.

On BBC Two's This Week, Ted Malloch was asked why he wanted to be US ambassador to the EU considering he is clearly not a fan of Brussels.

Mr Malloch replied: "I had in a previous career a diplomatic post where I helped bring down the Soviet Union. So maybe there's another union that needs a little taming."

These declarations didn't go unnoticed (source):

The liberal leader in the European Parliament, Guy Verhofstadt, and the leader of the European People’s Party, Manfred Weber, sent a letter to Juncker and European Council President Donald Tusk saying that Malloch should not be granted credentials, citing his “outrageous malevolence against the values that define this European Union.”

“We are strongly convinced that people seeing as their mission to disrupt or dissolve the European Union should not be accredited as official representatives to the EU,” the letter said.

Another reaction can be found here.

Question: who can actually decide if a person is fit to be a USA Ambassador for European Union?

Answer 1

Can the European Union refuse a proposed ambassador

Of course it can.

Vienna Convention

Proposed Ambassador

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-place Ambassador

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

for example

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

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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

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Accreditation Process

who decides

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

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Footnote

†

Similar to a state, the EU has a legislative branch (Parliament + Council), executive branch (Commission) and independent judiciary (Court of Justice).

(My emphasis)

Institutional Affairs. Retrieved from Europa.eu

Answer 2

It seems that the EU can indeed reject an ambassador, and there are various bodies of the EU which can formulate this rejection:

In a startling move that threatens a major diplomatic row, the leaders of the conservative, socialist and liberal groups in Brussels have written to the European commission and the European council, whose members represent the 28 EU states, to reject the appointment of Ted Malloch.
[...]
For a nominated ambassador to the EU to receive accreditation as head of a mission in Brussels, that person requires the approval of the commission and the European council along with the member states and the European External Action Service, which is akin to a foreign office.

Answer 3

Another answer already covered diplomatic credentials in international law but another part of the question is how the decision process on the side of the EU would look like. The last decades have seen an effort to create a common foreign policy but that's a competence member states are very reluctant to give up so there are special rules and procedures in this area.

Generally speaking, the EU's foreign policy is implemented by the European External Action Service under the authority of the High Representative of the Union for Foreign Affairs and Security Policy, currently Frederica Mogherini. The High representative is also automatically a vice president of the Commission, chairs the meetings of EU Foreign Ministers (the Foreign Affairs Council) and takes part in the meetings of the European Council so that she is the only person to seat in all potentially relevant meetings.

I could not find any clear rule or authoritative information about the reception of foreign diplomats but my guess is that such a politically sensitive decision would formally be taken at the highest level, either the Council of the EU (i.e. the Foreign Affairs Council) or the European Council (it's not the same thing). It would presumably be discussed beforehand in the COREPER II. I am not sure whether the president of the Commission could legally speaking refuse an ambassador without involving the Council but I don't see him doing that without prior agreement from the member states.

Conspicuously absent from this list of committees and institutions is the parliament, which has no role in this. As always, its members are making the most noise because they don't have any power to actually do anything about it.