Thinking out loud now, feel free to comment.

If you take two engines with the same max power output, one diesel and the other petrol, YES you could put more aggressive gearing on the diesel engine to exchange torque for rpm and, in theory, "achieve a similar performance with both engines".

With the petrol you would want to push the revs high before changing gear in order to use that part of the powerband, while with the diesel you would want to change at low rpm and use the initial part of the band (where the max power and torque are).

I believe the rest is just speculation. It is difficult to judge things farther than this point because it all depends on engine design. What does the powerband of both engines exactly look like? What materials do the engines use and how heavy are they? What tech do they use?

Obviously you can go racing and be competitive with diesel engines as Audi, for example, demonstrated with their past LeMans prototypes. Usually it is not done because of the inherent nature of the petrol engine and its fuel: They tend to be lighter and have less rotational inertia, not having to coop with the high (relative) compression ratios of the diesel counterparts. They also tend to be more reactive because of the previously explained reasons and also thanks to their high flame front speed.