Some good and some very good advice in this thread, all useful for others in the similar situation:

1. In the event of a misconnect due to a delay, it is the responsibility of the late-delivering carrier to handle the rebooking. That would be UA. Not LH. UA may well have LH handle the ticket reissue, but it is nonetheless UA's responsibility.

2. Do not start a Twitter-storm, have relatives contact UA and rearrange your itinerary, especially when connecting through a major hub such as FRA. What happened here is common. The inbound made up time enroute and the outbound took a short delay. End result is that OP made the connection. While it is possible that UA would have accepted a voluntary rebook had OP asked for it, the standard answer is to wait for FRA and deal with it there. OP's thread title was incorrect. This was not an "inevitable misconnect".

3. EC 261/2004 cancellation (delay) comepnsation would not have applied had UA's delay into FRA caused a delay resulting in a misconnect and a 3+ / 4+ hour delay at VCE. UA is only subject to the Regulation for departures from the EU. However, had LH [U]involuntarily[U] offloaded OP from its FRA-VCE flight on the assumption that he would misconnect when he did not, OP would have been entitled to denied boarding compensation from LH.

4. As a matter of good practice, it would be a good idea to have reroutes in hand when traveling. When IRROPS do occur, best to know what you want rather than to rely on carriers to have your best interest in mind. While FRA-VCE is not a worrisome reroute, it is good to know what the next flights are and to ask for those. It is also best to deal with compensation issues later and to focus on rerouting in the moment. In the moment, the goal is to get to VCE. Other than when you are in the middle of duty of care issues, no carrier's ground agents will debate the finer points of the Regulation with you and that simply causes your blood pressure to rise.