I had an egg crack in the carton and cover the other eggs with egg white. I consider the cracked egg a lost cause, but what about the splash damage? Are the eggs that have received an inadvertent egg white wash safe to eat? Particularly, for someone who may be very vulnerable to infection?
If the egg actually exploded as in the title, then no, that egg is probably highly contaminated. Not only is it bad, but nothing its contents have touched can be considered safe to use and should be discarded.
I assume however that you actually have an egg that was broken in handling. In that case, only the broken egg is a loss. Other answers and comments have addressed that, choice by location and your own personal sense of safe for you. I hate wasting food, but eggs today are a relatively inexpensive commodity for most of us, so if you feel uneasy, error on the side of safety.
When an egg is only partially cracked, but the membrane is intact, that egg is normally safe for use for several days. You have lost some protection, but not the immediate integrity of the egg. One issue though is you do not know when it was cracked. In your possession can be safe, my personal rule being about a week. But if it already was cracked, I do not know if it happened before washing (in the US) or after. If before, the egg is compromised. Now, this is one spot that the EU with non-washed eggs are at a slight, very slight, disadvantage. IMO, unwashed, a cracked egg is compromised even if the membrane is intact. That paragraph is not exactly your question, but could be for people searching that find their way here.
Edit after the question was substantially changed. This answer wasn't initially concerned with the fate of the broken egg itself, but the others around it.
If the white 'exploded' bin it. If it 'leaked' then that egg is compromised. If you broke it, eat it today. If you don't know when it was broken, discard & treat the rest of this answer as it stood before you changed the question.
Don't wash European eggs.
US eggs already have had their natural protective layer washed off, so this action would be sensible to prevent further contamination.
EU eggs still have their natural protection, so should be left alone. Wash right before use if you're worried.
I am new to your site so I may not understand how to view all the answers or the entire discussion, but I did not see the solution my mother taught me. You can use the same "trick" to test for freshness that you can use to test for potential contamination in your egg, so I will give the answers to both. (I noticed some people do not like you deviating from the original question but, hey, I'm old. I figure it is NEVER too late to learn something that may come in handy some day.)
Place your egg in a cup or bowl of cold water - cold tap water, not cold as in refrigerated.
Make sure there is enough water that the egg can stand up on end in the container and still be covered with water.
If the submerged egg lies flat along the bottom, the egg is very fresh. ENJOY! If it doesn't but was sold as fresh, you might want to mention that to the vendor.
If the egg stands up on one end it is not so fresh, up to maybe a few weeks, but still good to eat.
If the egg floats, it contains air and should be tossed. It is not only old, but air can contain contaminants and air is how contaminants get into your egg.
Believe it or not, I have had eggs that have been in the refrigerator for months that stand on end in water and are just fine. But remember ALL pre-washed eggs can contain air and contaminants that can cause Salmonella-and-such so always thoroughly cook your eggs. If you must drop a raw egg in a drink, you might want to make sure it lays flat on the bottom of a glass of water first.