It's possible the result could have been different, but given the voting system used in the UK for European elections, it appears it would have taken a large number of additional votes, very specifically arranged to cause it to happen.
Taking the results from this wikipedia page and applying the D'Hondt system of PR to the votes in each electoral region we find :
- In the South East of England the 9th seat went to the Labour party with a quota of 184,678 votes (which was their total share, this was their first seat). Meanwhile Change UK received no seats for 105,832. Hence about 80,000 would have been enough.
- In London, a very similar total would have flipped a Brexit party seat to Change UK.
- In the East Midlands about 26,000 votes would have flipped a Brexit Party seat to the Greens
- In the West Midlands about 50,000 votes would have flipped a Conservative seat to the Liberal Democrats.
- In Yorkshire 96,000 votes would have flipped a seat from the Brexit party to the Liberal Democrats
I'm afraid I don't understand Northern Ireland politics well enough to try and work out what would need to happen under their STV system. In Scotland about 36,000 votes would have flipped a seat from the Conservatives to Labour, but I'm not sure that's actually a pro-Remain move.