No, IPsec cannot and will not replace HTTPS because they are effectively Apples and Oranges, here's why:
IPsec operates at the network layer and is negotiated explicitly via IKE - both endpoints have to agree on a keying scheme, mode of operation and a number of other parameters which are then installed into whatever is responsible for the IP(v6) stack (typically the kernel of the operating system).
*IPsec is designed for protecting an entire channel (or channels) of communication over a prolonged period as opposed to simply setting up a brief session, exchanging data and tearing it down.
Hence, IPsec simply does not scale well for the sort of lifetimes that HTTPS typically experiences (i.e. repetitive, short-lived connections), not to mention the fact that it explicitly relies on the IP stack itself makes implementing it within the application both problematic and a security issue, since configuration of IPsec parameters are generally a privileged operation requiring root/administrator access.
HTTPS on the other hand, more specifically, SSL/TLS operates on the basis of the application (such as your web browser) having a pre-defined list of authorities that it trusts to sign server encryption certificates.
As mentioned in another answer, whilst DNSSEC could very well be used to bootstrap IPsec, This is already perfectly doable for TLS and indeed has been codified within the IETF as RFC6698.